Main:z from

Percentage Accurate: 91.2% → 97.7%

Time: 43.4s

Alternatives: 21

Speedup: 1.5×

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: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(\left(t\_2 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{z} - t\_3\right)\\ \mathbf{if}\;t\_4 \leq 1.00005:\\ \;\;\;\;\frac{1}{t\_2 + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \left(t\_1 + \left(\left(t\_3 - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(t\_1 - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4
         (-
          (+ (- t_2 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt z) t_3))))
   (if (<= t_4 1.00005)
     (+
      (/ 1.0 (+ t_2 (sqrt x)))
      (+ (* 0.5 (sqrt (/ 1.0 y))) (+ t_1 (- (- t_3 (sqrt z)) (sqrt t)))))
     (+ t_4 (- t_1 (sqrt t))))))

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));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = ((t_2 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) - (sqrt(z) - t_3);
	double tmp;
	if (t_4 <= 1.00005) {
		tmp = (1.0 / (t_2 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_1 + ((t_3 - sqrt(z)) - sqrt(t))));
	} else {
		tmp = t_4 + (t_1 - sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    t_4 = ((t_2 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) - (sqrt(z) - t_3)
    if (t_4 <= 1.00005d0) then
        tmp = (1.0d0 / (t_2 + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) + (t_1 + ((t_3 - sqrt(z)) - sqrt(t))))
    else
        tmp = t_4 + (t_1 - 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((t + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = ((t_2 - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) - (Math.sqrt(z) - t_3);
	double tmp;
	if (t_4 <= 1.00005) {
		tmp = (1.0 / (t_2 + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) + (t_1 + ((t_3 - Math.sqrt(z)) - Math.sqrt(t))));
	} else {
		tmp = t_4 + (t_1 - 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((t + 1.0))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((z + 1.0))
	t_4 = ((t_2 - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) - (math.sqrt(z) - t_3)
	tmp = 0
	if t_4 <= 1.00005:
		tmp = (1.0 / (t_2 + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) + (t_1 + ((t_3 - math.sqrt(z)) - math.sqrt(t))))
	else:
		tmp = t_4 + (t_1 - 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(t + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) - Float64(sqrt(z) - t_3))
	tmp = 0.0
	if (t_4 <= 1.00005)
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(t_1 + Float64(Float64(t_3 - sqrt(z)) - sqrt(t)))));
	else
		tmp = Float64(t_4 + Float64(t_1 - 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((t + 1.0));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((z + 1.0));
	t_4 = ((t_2 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) - (sqrt(z) - t_3);
	tmp = 0.0;
	if (t_4 <= 1.00005)
		tmp = (1.0 / (t_2 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_1 + ((t_3 - sqrt(z)) - sqrt(t))));
	else
		tmp = t_4 + (t_1 - 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[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.00005], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. associate-+l- 79.1%

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

      6. +-commutative 79.1%

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

      7. +-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. flip-- 79.1%

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

      2. add-sqr-sqrt 48.2%

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

      3. +-commutative 48.2%

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

      4. add-sqr-sqrt 79.3%

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

      5. +-commutative 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. associate--l+ 81.1%

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

      2. +-inverses 81.1%

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

      3. metadata-eval 81.1%

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

   8. Simplified 81.1%

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

   9. Taylor expanded in y around inf 65.8%

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

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 1.00005:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{t + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{t + 1}\\ \mathbf{if}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{z + 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(t\_2 - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(t\_1 - \sqrt{y}\right) - \left(\sqrt{t} - t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ t 1.0))))
   (if (<= z 2.9e+21)
     (+
      t_1
      (+
       (+ (sqrt (+ z 1.0)) (- (- (sqrt (+ x 1.0)) (sqrt x)) (sqrt z)))
       (- (- t_2 (sqrt t)) (sqrt y))))
     (+
      (/ 1.0 (+ (hypot 1.0 (sqrt x)) (sqrt x)))
      (- (- t_1 (sqrt y)) (- (sqrt t) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((t + 1.0));
	double tmp;
	if (z <= 2.9e+21) {
		tmp = t_1 + ((sqrt((z + 1.0)) + ((sqrt((x + 1.0)) - sqrt(x)) - sqrt(z))) + ((t_2 - sqrt(t)) - sqrt(y)));
	} else {
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((t_1 - sqrt(y)) - (sqrt(t) - t_2));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((t + 1.0));
	double tmp;
	if (z <= 2.9e+21) {
		tmp = t_1 + ((Math.sqrt((z + 1.0)) + ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) - Math.sqrt(z))) + ((t_2 - Math.sqrt(t)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.hypot(1.0, Math.sqrt(x)) + Math.sqrt(x))) + ((t_1 - Math.sqrt(y)) - (Math.sqrt(t) - t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((t + 1.0))
	tmp = 0
	if z <= 2.9e+21:
		tmp = t_1 + ((math.sqrt((z + 1.0)) + ((math.sqrt((x + 1.0)) - math.sqrt(x)) - math.sqrt(z))) + ((t_2 - math.sqrt(t)) - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.hypot(1.0, math.sqrt(x)) + math.sqrt(x))) + ((t_1 - math.sqrt(y)) - (math.sqrt(t) - t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (z <= 2.9e+21)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(z + 1.0)) + Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) - sqrt(z))) + Float64(Float64(t_2 - sqrt(t)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, sqrt(x)) + sqrt(x))) + Float64(Float64(t_1 - sqrt(y)) - Float64(sqrt(t) - 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((y + 1.0));
	t_2 = sqrt((t + 1.0));
	tmp = 0.0;
	if (z <= 2.9e+21)
		tmp = t_1 + ((sqrt((z + 1.0)) + ((sqrt((x + 1.0)) - sqrt(x)) - sqrt(z))) + ((t_2 - sqrt(t)) - sqrt(y)));
	else
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((t_1 - sqrt(y)) - (sqrt(t) - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.9e+21], N[(t$95$1 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.9e21

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 73.9%

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

      4. associate-+l- 62.2%

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

      5. associate-+r- 47.1%

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

   3. Simplified 47.1%

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

   if 2.9e21 < z

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. flip-- 87.6%

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

      2. add-sqr-sqrt 63.9%

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

      3. +-commutative 63.9%

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

      4. add-sqr-sqrt 87.8%

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

      5. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. hypot-1-def 89.3%

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

   10. Applied egg-rr 89.3%

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

   11. Taylor expanded in z around inf 89.3%

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

       1. +-commutative 89.3%

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

   13. Simplified 89.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\sqrt{z + 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(t\_1 - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - t\_1\right)\right)\\ \end{array} \end{array} \]

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))))
   (if (<= z 5.4e-34)
     (+ 3.0 (- t_1 (+ (+ (sqrt y) (sqrt z)) (+ (sqrt t) (sqrt x)))))
     (if (<= z 2.9e+21)
       (+
        (- 1.0 (sqrt x))
        (+
         (- 1.0 (sqrt y))
         (+ (sqrt (+ z 1.0)) (- (* 0.5 (sqrt (/ 1.0 t))) (sqrt z)))))
       (+
        (/ 1.0 (+ (hypot 1.0 (sqrt x)) (sqrt x)))
        (- (- (sqrt (+ y 1.0)) (sqrt y)) (- (sqrt t) 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));
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (t_1 - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	} else if (z <= 2.9e+21) {
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	} else {
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((sqrt((y + 1.0)) - sqrt(y)) - (sqrt(t) - t_1));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0));
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (t_1 - ((Math.sqrt(y) + Math.sqrt(z)) + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 2.9e+21) {
		tmp = (1.0 - Math.sqrt(x)) + ((1.0 - Math.sqrt(y)) + (Math.sqrt((z + 1.0)) + ((0.5 * Math.sqrt((1.0 / t))) - Math.sqrt(z))));
	} else {
		tmp = (1.0 / (Math.hypot(1.0, Math.sqrt(x)) + Math.sqrt(x))) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) - (Math.sqrt(t) - 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))
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (t_1 - ((math.sqrt(y) + math.sqrt(z)) + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 2.9e+21:
		tmp = (1.0 - math.sqrt(x)) + ((1.0 - math.sqrt(y)) + (math.sqrt((z + 1.0)) + ((0.5 * math.sqrt((1.0 / t))) - math.sqrt(z))))
	else:
		tmp = (1.0 / (math.hypot(1.0, math.sqrt(x)) + math.sqrt(x))) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) - (math.sqrt(t) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(t_1 - Float64(Float64(sqrt(y) + sqrt(z)) + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 2.9e+21)
		tmp = Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(1.0 - sqrt(y)) + Float64(sqrt(Float64(z + 1.0)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) - sqrt(z)))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, sqrt(x)) + sqrt(x))) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) - Float64(sqrt(t) - 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));
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (t_1 - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	elseif (z <= 2.9e+21)
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	else
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((sqrt((y + 1.0)) - sqrt(y)) - (sqrt(t) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.4e-34], N[(3.0 + N[(t$95$1 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e+21], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 2.9e21

   1. Initial program 91.2%

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

      1. associate-+l+ 91.2%

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

      2. associate-+l+ 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+l- 54.5%

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

      6. +-commutative 54.5%

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

      7. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in y around 0 19.0%

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

   6. Taylor expanded in x around 0 7.8%

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

   7. Taylor expanded in t around inf 12.0%

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

      1. associate--l+ 12.0%

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

   9. Simplified 12.0%

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

   if 2.9e21 < z

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. flip-- 87.6%

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

      2. add-sqr-sqrt 63.9%

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

      3. +-commutative 63.9%

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

      4. add-sqr-sqrt 87.8%

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

      5. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. hypot-1-def 89.3%

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

   10. Applied egg-rr 89.3%

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

   11. Taylor expanded in z around inf 89.3%

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

       1. +-commutative 89.3%

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

   13. Simplified 89.3%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;t\_2 + \left(\left(\left(\sqrt{z + 1} + \left(1 - \sqrt{x}\right)\right) - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(t\_2 - \sqrt{y}\right) - \left(\sqrt{t} - t\_1\right)\right)\\ \end{array} \end{array} \]

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))) (t_2 (sqrt (+ y 1.0))))
   (if (<= z 2.9e+21)
     (+
      t_2
      (+
       (- (+ (sqrt (+ z 1.0)) (- 1.0 (sqrt x))) (sqrt z))
       (- (- t_1 (sqrt t)) (sqrt y))))
     (+
      (/ 1.0 (+ (hypot 1.0 (sqrt x)) (sqrt x)))
      (- (- t_2 (sqrt y)) (- (sqrt t) 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));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (z <= 2.9e+21) {
		tmp = t_2 + (((sqrt((z + 1.0)) + (1.0 - sqrt(x))) - sqrt(z)) + ((t_1 - sqrt(t)) - sqrt(y)));
	} else {
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((t_2 - sqrt(y)) - (sqrt(t) - t_1));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 2.9e+21) {
		tmp = t_2 + (((Math.sqrt((z + 1.0)) + (1.0 - Math.sqrt(x))) - Math.sqrt(z)) + ((t_1 - Math.sqrt(t)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.hypot(1.0, Math.sqrt(x)) + Math.sqrt(x))) + ((t_2 - Math.sqrt(y)) - (Math.sqrt(t) - 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))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 2.9e+21:
		tmp = t_2 + (((math.sqrt((z + 1.0)) + (1.0 - math.sqrt(x))) - math.sqrt(z)) + ((t_1 - math.sqrt(t)) - math.sqrt(y)))
	else:
		tmp = (1.0 / (math.hypot(1.0, math.sqrt(x)) + math.sqrt(x))) + ((t_2 - math.sqrt(y)) - (math.sqrt(t) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 2.9e+21)
		tmp = Float64(t_2 + Float64(Float64(Float64(sqrt(Float64(z + 1.0)) + Float64(1.0 - sqrt(x))) - sqrt(z)) + Float64(Float64(t_1 - sqrt(t)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, sqrt(x)) + sqrt(x))) + Float64(Float64(t_2 - sqrt(y)) - Float64(sqrt(t) - 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));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 2.9e+21)
		tmp = t_2 + (((sqrt((z + 1.0)) + (1.0 - sqrt(x))) - sqrt(z)) + ((t_1 - sqrt(t)) - sqrt(y)));
	else
		tmp = (1.0 / (hypot(1.0, sqrt(x)) + sqrt(x))) + ((t_2 - sqrt(y)) - (sqrt(t) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 2.9e+21], N[(t$95$2 + N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.9e21

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 73.9%

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

      4. associate-+l- 62.2%

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

      5. associate-+r- 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in x around 0 23.2%

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

      1. associate--r+ 23.2%

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

      2. +-commutative 23.2%

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

      3. associate--l+ 23.2%

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

   7. Simplified 23.2%

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

   if 2.9e21 < z

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. flip-- 87.6%

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

      2. add-sqr-sqrt 63.9%

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

      3. +-commutative 63.9%

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

      4. add-sqr-sqrt 87.8%

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

      5. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. hypot-1-def 89.3%

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

   10. Applied egg-rr 89.3%

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

   11. Taylor expanded in z around inf 89.3%

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

       1. +-commutative 89.3%

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

   13. Simplified 89.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{y + 1} + \left(\left(\left(\sqrt{z + 1} + \left(1 - \sqrt{x}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(t\_1 - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - t\_1\right)\right)\\ \end{array} \end{array} \]

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))))
   (if (<= z 5.4e-34)
     (+ 3.0 (- t_1 (+ (+ (sqrt y) (sqrt z)) (+ (sqrt t) (sqrt x)))))
     (if (<= z 2.9e+21)
       (+
        (- 1.0 (sqrt x))
        (+
         (- 1.0 (sqrt y))
         (+ (sqrt (+ z 1.0)) (- (* 0.5 (sqrt (/ 1.0 t))) (sqrt z)))))
       (+
        (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
        (- (- (sqrt (+ y 1.0)) (sqrt y)) (- (sqrt t) 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));
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (t_1 - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	} else if (z <= 2.9e+21) {
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((sqrt((y + 1.0)) - sqrt(y)) - (sqrt(t) - t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0))
    if (z <= 5.4d-34) then
        tmp = 3.0d0 + (t_1 - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))))
    else if (z <= 2.9d+21) then
        tmp = (1.0d0 - sqrt(x)) + ((1.0d0 - sqrt(y)) + (sqrt((z + 1.0d0)) + ((0.5d0 * sqrt((1.0d0 / t))) - sqrt(z))))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((sqrt((y + 1.0d0)) - sqrt(y)) - (sqrt(t) - 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));
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (t_1 - ((Math.sqrt(y) + Math.sqrt(z)) + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 2.9e+21) {
		tmp = (1.0 - Math.sqrt(x)) + ((1.0 - Math.sqrt(y)) + (Math.sqrt((z + 1.0)) + ((0.5 * Math.sqrt((1.0 / t))) - Math.sqrt(z))));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) - (Math.sqrt(t) - 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))
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (t_1 - ((math.sqrt(y) + math.sqrt(z)) + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 2.9e+21:
		tmp = (1.0 - math.sqrt(x)) + ((1.0 - math.sqrt(y)) + (math.sqrt((z + 1.0)) + ((0.5 * math.sqrt((1.0 / t))) - math.sqrt(z))))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) - (math.sqrt(t) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(t_1 - Float64(Float64(sqrt(y) + sqrt(z)) + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 2.9e+21)
		tmp = Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(1.0 - sqrt(y)) + Float64(sqrt(Float64(z + 1.0)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) - sqrt(z)))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) - Float64(sqrt(t) - 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));
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (t_1 - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	elseif (z <= 2.9e+21)
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((sqrt((y + 1.0)) - sqrt(y)) - (sqrt(t) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.4e-34], N[(3.0 + N[(t$95$1 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e+21], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 2.9e21

   1. Initial program 91.2%

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

      1. associate-+l+ 91.2%

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

      2. associate-+l+ 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+l- 54.5%

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

      6. +-commutative 54.5%

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

      7. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in y around 0 19.0%

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

   6. Taylor expanded in x around 0 7.8%

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

   7. Taylor expanded in t around inf 12.0%

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

      1. associate--l+ 12.0%

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

   9. Simplified 12.0%

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

   if 2.9e21 < z

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. flip-- 87.6%

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

      2. add-sqr-sqrt 63.9%

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

      3. +-commutative 63.9%

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

      4. add-sqr-sqrt 87.8%

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

      5. +-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 89.3%

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

   9. Taylor expanded in z around inf 89.3%

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

       1. +-commutative 89.3%

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

   11. Simplified 89.3%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+21}:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 490000:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + x \cdot \left(\frac{\sqrt{y + 1} - \sqrt{y}}{x} - \sqrt{\frac{1}{x}}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.4e-34)
   (+ 3.0 (- (sqrt (+ t 1.0)) (+ (+ (sqrt y) (sqrt z)) (+ (sqrt t) (sqrt x)))))
   (if (<= z 490000.0)
     (+
      (- 1.0 (sqrt x))
      (+
       (- 1.0 (sqrt y))
       (+ (sqrt (+ z 1.0)) (- (* 0.5 (sqrt (/ 1.0 t))) (sqrt z)))))
     (+
      (+
       (sqrt (+ x 1.0))
       (* x (- (/ (- (sqrt (+ y 1.0)) (sqrt y)) x) (sqrt (/ 1.0 x)))))
      (* 0.5 (sqrt (/ 1.0 z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (sqrt((t + 1.0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	} else if (z <= 490000.0) {
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	} else {
		tmp = (sqrt((x + 1.0)) + (x * (((sqrt((y + 1.0)) - sqrt(y)) / x) - sqrt((1.0 / x))))) + (0.5 * sqrt((1.0 / z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5.4d-34) then
        tmp = 3.0d0 + (sqrt((t + 1.0d0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))))
    else if (z <= 490000.0d0) then
        tmp = (1.0d0 - sqrt(x)) + ((1.0d0 - sqrt(y)) + (sqrt((z + 1.0d0)) + ((0.5d0 * sqrt((1.0d0 / t))) - sqrt(z))))
    else
        tmp = (sqrt((x + 1.0d0)) + (x * (((sqrt((y + 1.0d0)) - sqrt(y)) / x) - sqrt((1.0d0 / x))))) + (0.5d0 * sqrt((1.0d0 / z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (Math.sqrt((t + 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 490000.0) {
		tmp = (1.0 - Math.sqrt(x)) + ((1.0 - Math.sqrt(y)) + (Math.sqrt((z + 1.0)) + ((0.5 * Math.sqrt((1.0 / t))) - Math.sqrt(z))));
	} else {
		tmp = (Math.sqrt((x + 1.0)) + (x * (((Math.sqrt((y + 1.0)) - Math.sqrt(y)) / x) - Math.sqrt((1.0 / x))))) + (0.5 * Math.sqrt((1.0 / z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (math.sqrt((t + 1.0)) - ((math.sqrt(y) + math.sqrt(z)) + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 490000.0:
		tmp = (1.0 - math.sqrt(x)) + ((1.0 - math.sqrt(y)) + (math.sqrt((z + 1.0)) + ((0.5 * math.sqrt((1.0 / t))) - math.sqrt(z))))
	else:
		tmp = (math.sqrt((x + 1.0)) + (x * (((math.sqrt((y + 1.0)) - math.sqrt(y)) / x) - math.sqrt((1.0 / x))))) + (0.5 * math.sqrt((1.0 / z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(sqrt(Float64(t + 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 490000.0)
		tmp = Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(1.0 - sqrt(y)) + Float64(sqrt(Float64(z + 1.0)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) - sqrt(z)))));
	else
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(x * Float64(Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) / x) - sqrt(Float64(1.0 / x))))) + Float64(0.5 * sqrt(Float64(1.0 / z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (sqrt((t + 1.0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	elseif (z <= 490000.0)
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	else
		tmp = (sqrt((x + 1.0)) + (x * (((sqrt((y + 1.0)) - sqrt(y)) / x) - sqrt((1.0 / x))))) + (0.5 * sqrt((1.0 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 5.4e-34], N[(3.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 490000.0], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(x * N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 4.9e5

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. associate-+l+ 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+l- 52.8%

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

      6. +-commutative 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around 0 20.9%

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

   6. Taylor expanded in x around 0 9.3%

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

   7. Taylor expanded in t around inf 14.3%

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

      1. associate--l+ 14.3%

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

   9. Simplified 14.3%

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

   if 4.9e5 < 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. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-+r- 70.1%

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

      4. associate-+l- 51.7%

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

      5. associate-+r- 51.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.5%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. distribute-rgt-neg-in 0.0%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in z around inf 27.3%

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

       1. associate-+r+ 30.5%

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

       2. mul-1-neg 30.5%

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

       3. unsub-neg 30.5%

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

       4. mul-1-neg 30.5%

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

       5. unsub-neg 30.5%

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

   13. Simplified 30.5%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 490000:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + x \cdot \left(\frac{\sqrt{y + 1} - \sqrt{y}}{x} - \sqrt{\frac{1}{x}}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1250000:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 5.4e-34)
   (+ 3.0 (- (sqrt (+ t 1.0)) (+ (+ (sqrt y) (sqrt z)) (+ (sqrt t) (sqrt x)))))
   (if (<= z 1250000.0)
     (+
      (- 1.0 (sqrt x))
      (+
       (- 1.0 (sqrt y))
       (+ (sqrt (+ z 1.0)) (- (* 0.5 (sqrt (/ 1.0 t))) (sqrt z)))))
     (+
      (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x)))
      1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (sqrt((t + 1.0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	} else if (z <= 1250000.0) {
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	} else {
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5.4d-34) then
        tmp = 3.0d0 + (sqrt((t + 1.0d0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))))
    else if (z <= 1250000.0d0) then
        tmp = (1.0d0 - sqrt(x)) + ((1.0d0 - sqrt(y)) + (sqrt((z + 1.0d0)) + ((0.5d0 * sqrt((1.0d0 / t))) - sqrt(z))))
    else
        tmp = ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (Math.sqrt((t + 1.0)) - ((Math.sqrt(y) + Math.sqrt(z)) + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 1250000.0) {
		tmp = (1.0 - Math.sqrt(x)) + ((1.0 - Math.sqrt(y)) + (Math.sqrt((z + 1.0)) + ((0.5 * Math.sqrt((1.0 / t))) - Math.sqrt(z))));
	} else {
		tmp = ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (math.sqrt((t + 1.0)) - ((math.sqrt(y) + math.sqrt(z)) + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 1250000.0:
		tmp = (1.0 - math.sqrt(x)) + ((1.0 - math.sqrt(y)) + (math.sqrt((z + 1.0)) + ((0.5 * math.sqrt((1.0 / t))) - math.sqrt(z))))
	else:
		tmp = ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(sqrt(Float64(t + 1.0)) - Float64(Float64(sqrt(y) + sqrt(z)) + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 1250000.0)
		tmp = Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(1.0 - sqrt(y)) + Float64(sqrt(Float64(z + 1.0)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) - sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (sqrt((t + 1.0)) - ((sqrt(y) + sqrt(z)) + (sqrt(t) + sqrt(x))));
	elseif (z <= 1250000.0)
		tmp = (1.0 - sqrt(x)) + ((1.0 - sqrt(y)) + (sqrt((z + 1.0)) + ((0.5 * sqrt((1.0 / t))) - sqrt(z))));
	else
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 5.4e-34], N[(3.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1250000.0], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 1.25e6

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. associate-+l+ 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+l- 52.8%

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

      6. +-commutative 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around 0 20.9%

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

   6. Taylor expanded in x around 0 9.3%

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

   7. Taylor expanded in t around inf 14.3%

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

      1. associate--l+ 14.3%

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

   9. Simplified 14.3%

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

   if 1.25e6 < 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. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-+r- 70.1%

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

      4. associate-+l- 51.7%

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

      5. associate-+r- 51.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.5%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in z around inf 27.0%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in x around 0 13.8%

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

       1. associate--l+ 32.5%

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

   13. Simplified 32.5%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1250000:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \left(\left(1 - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(0.5 \cdot \sqrt{\frac{1}{t}} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{z}\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(t\_1 + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 500000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} + 0.5 \cdot \sqrt{\frac{1}{t}}\right) - \left(t\_1 + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 y) (sqrt z))))
   (if (<= z 5.4e-34)
     (+ 3.0 (- (sqrt (+ t 1.0)) (+ t_1 (+ (sqrt t) (sqrt x)))))
     (if (<= z 500000.0)
       (+
        2.0
        (- (+ (sqrt (+ z 1.0)) (* 0.5 (sqrt (/ 1.0 t)))) (+ t_1 (sqrt x))))
       (+
        (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(z);
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (sqrt((t + 1.0)) - (t_1 + (sqrt(t) + sqrt(x))));
	} else if (z <= 500000.0) {
		tmp = 2.0 + ((sqrt((z + 1.0)) + (0.5 * sqrt((1.0 / t)))) - (t_1 + sqrt(x)));
	} else {
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(y) + sqrt(z)
    if (z <= 5.4d-34) then
        tmp = 3.0d0 + (sqrt((t + 1.0d0)) - (t_1 + (sqrt(t) + sqrt(x))))
    else if (z <= 500000.0d0) then
        tmp = 2.0d0 + ((sqrt((z + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / t)))) - (t_1 + sqrt(x)))
    else
        tmp = ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(y) + Math.sqrt(z);
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (Math.sqrt((t + 1.0)) - (t_1 + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 500000.0) {
		tmp = 2.0 + ((Math.sqrt((z + 1.0)) + (0.5 * Math.sqrt((1.0 / t)))) - (t_1 + Math.sqrt(x)));
	} else {
		tmp = ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt(z)
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (math.sqrt((t + 1.0)) - (t_1 + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 500000.0:
		tmp = 2.0 + ((math.sqrt((z + 1.0)) + (0.5 * math.sqrt((1.0 / t)))) - (t_1 + math.sqrt(x)))
	else:
		tmp = ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(z))
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(sqrt(Float64(t + 1.0)) - Float64(t_1 + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 500000.0)
		tmp = Float64(2.0 + Float64(Float64(sqrt(Float64(z + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / t)))) - Float64(t_1 + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt(z);
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (sqrt((t + 1.0)) - (t_1 + (sqrt(t) + sqrt(x))));
	elseif (z <= 500000.0)
		tmp = 2.0 + ((sqrt((z + 1.0)) + (0.5 * sqrt((1.0 / t)))) - (t_1 + sqrt(x)));
	else
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.4e-34], N[(3.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$1 + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 500000.0], N[(2.0 + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 5e5

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. associate-+l+ 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+l- 52.8%

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

      6. +-commutative 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around 0 20.9%

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

   6. Taylor expanded in x around 0 9.3%

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

   7. Taylor expanded in t around inf 14.3%

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

      1. associate--l+ 14.3%

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

   9. Simplified 14.3%

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

   if 5e5 < 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. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-+r- 70.1%

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

      4. associate-+l- 51.7%

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

      5. associate-+r- 51.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.5%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in z around inf 27.0%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in x around 0 13.8%

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

       1. associate--l+ 32.5%

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

   13. Simplified 32.5%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 500000:\\ \;\;\;\;2 + \left(\left(\sqrt{z + 1} + 0.5 \cdot \sqrt{\frac{1}{t}}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{z}\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(t\_1 + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 6000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(t\_1 + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 y) (sqrt z))))
   (if (<= z 5.4e-34)
     (+ 3.0 (- (sqrt (+ t 1.0)) (+ t_1 (+ (sqrt t) (sqrt x)))))
     (if (<= z 6000000.0)
       (- (+ (sqrt (+ z 1.0)) 2.0) (+ t_1 (sqrt x)))
       (+
        (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(z);
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (sqrt((t + 1.0)) - (t_1 + (sqrt(t) + sqrt(x))));
	} else if (z <= 6000000.0) {
		tmp = (sqrt((z + 1.0)) + 2.0) - (t_1 + sqrt(x));
	} else {
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(y) + sqrt(z)
    if (z <= 5.4d-34) then
        tmp = 3.0d0 + (sqrt((t + 1.0d0)) - (t_1 + (sqrt(t) + sqrt(x))))
    else if (z <= 6000000.0d0) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - (t_1 + sqrt(x))
    else
        tmp = ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(y) + Math.sqrt(z);
	double tmp;
	if (z <= 5.4e-34) {
		tmp = 3.0 + (Math.sqrt((t + 1.0)) - (t_1 + (Math.sqrt(t) + Math.sqrt(x))));
	} else if (z <= 6000000.0) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - (t_1 + Math.sqrt(x));
	} else {
		tmp = ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt(z)
	tmp = 0
	if z <= 5.4e-34:
		tmp = 3.0 + (math.sqrt((t + 1.0)) - (t_1 + (math.sqrt(t) + math.sqrt(x))))
	elif z <= 6000000.0:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - (t_1 + math.sqrt(x))
	else:
		tmp = ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(z))
	tmp = 0.0
	if (z <= 5.4e-34)
		tmp = Float64(3.0 + Float64(sqrt(Float64(t + 1.0)) - Float64(t_1 + Float64(sqrt(t) + sqrt(x)))));
	elseif (z <= 6000000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(t_1 + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt(z);
	tmp = 0.0;
	if (z <= 5.4e-34)
		tmp = 3.0 + (sqrt((t + 1.0)) - (t_1 + (sqrt(t) + sqrt(x))));
	elseif (z <= 6000000.0)
		tmp = (sqrt((z + 1.0)) + 2.0) - (t_1 + sqrt(x));
	else
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.4e-34], N[(3.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$1 + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6000000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.40000000000000034e-34

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 65.2%

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

      6. +-commutative 65.2%

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

      7. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in y around 0 30.7%

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

   6. Taylor expanded in x around 0 15.1%

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

   7. Taylor expanded in z around 0 14.0%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 22.8%

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

      2. associate-+r+ 22.8%

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

   9. Simplified 22.8%

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

   if 5.40000000000000034e-34 < z < 6e6

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. associate-+l+ 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+l- 52.8%

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

      6. +-commutative 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around 0 20.9%

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

   6. Taylor expanded in x around 0 9.3%

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

   7. Taylor expanded in t around inf 13.9%

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

   if 6e6 < 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. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-+r- 70.1%

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

      4. associate-+l- 51.7%

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

      5. associate-+r- 51.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.5%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in z around inf 27.0%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in x around 0 13.8%

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

       1. associate--l+ 32.5%

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

   13. Simplified 32.5%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;3 + \left(\sqrt{t + 1} - \left(\left(\sqrt{y} + \sqrt{z}\right) + \left(\sqrt{t} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 6000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 70000000:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + t\_1\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 z))))
   (if (<= y 6.6e-24)
     (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt z) (sqrt x))))
     (if (<= y 70000000.0)
       (+ (- (+ (sqrt (+ y 1.0)) (* 0.5 t_1)) (+ (sqrt y) (sqrt x))) 1.0)
       (+ (sqrt (+ x 1.0)) (- (* 0.5 (+ (sqrt (/ 1.0 y)) t_1)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double tmp;
	if (y <= 6.6e-24) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(x)));
	} else if (y <= 70000000.0) {
		tmp = ((sqrt((y + 1.0)) + (0.5 * t_1)) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 / z))
    if (y <= 6.6d-24) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(z) + sqrt(x)))
    else if (y <= 70000000.0d0) then
        tmp = ((sqrt((y + 1.0d0)) + (0.5d0 * t_1)) - (sqrt(y) + sqrt(x))) + 1.0d0
    else
        tmp = sqrt((x + 1.0d0)) + ((0.5d0 * (sqrt((1.0d0 / y)) + t_1)) - sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 / z));
	double tmp;
	if (y <= 6.6e-24) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + Math.sqrt(x)));
	} else if (y <= 70000000.0) {
		tmp = ((Math.sqrt((y + 1.0)) + (0.5 * t_1)) - (Math.sqrt(y) + Math.sqrt(x))) + 1.0;
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((0.5 * (Math.sqrt((1.0 / y)) + t_1)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 / z))
	tmp = 0
	if y <= 6.6e-24:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(z) + math.sqrt(x)))
	elif y <= 70000000.0:
		tmp = ((math.sqrt((y + 1.0)) + (0.5 * t_1)) - (math.sqrt(y) + math.sqrt(x))) + 1.0
	else:
		tmp = math.sqrt((x + 1.0)) + ((0.5 * (math.sqrt((1.0 / y)) + t_1)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (y <= 6.6e-24)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + sqrt(x))));
	elseif (y <= 70000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * t_1)) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(0.5 * Float64(sqrt(Float64(1.0 / y)) + t_1)) - sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 / z));
	tmp = 0.0;
	if (y <= 6.6e-24)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(x)));
	elseif (y <= 70000000.0)
		tmp = ((sqrt((y + 1.0)) + (0.5 * t_1)) - (sqrt(y) + sqrt(x))) + 1.0;
	else
		tmp = sqrt((x + 1.0)) + ((0.5 * (sqrt((1.0 / y)) + t_1)) - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6.6e-24], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 70000000.0], N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.59999999999999968e-24

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-+r+ 98.2%

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

      3. associate-+r- 98.2%

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

      4. associate-+l- 98.2%

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

      5. associate-+r- 98.2%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in t around inf 18.2%

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

      1. associate--l+ 22.5%

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

      2. +-commutative 22.5%

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

      3. +-commutative 22.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in z around inf 22.5%

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

   9. Taylor expanded in x around 0 26.2%

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

   10. Taylor expanded in y around 0 15.4%

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

       1. associate--l+ 28.2%

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

   12. Simplified 28.2%

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

   if 6.59999999999999968e-24 < y < 7e7

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. associate-+r+ 90.4%

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

      3. associate-+r- 90.3%

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

      4. associate-+l- 90.3%

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

      5. associate-+r- 90.3%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in t around inf 10.6%

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

      1. associate--l+ 17.8%

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

      2. +-commutative 17.8%

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

      3. +-commutative 17.8%

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

   7. Simplified 17.8%

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

   8. Taylor expanded in z around inf 28.4%

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

      1. associate--l+ 28.4%

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

      2. +-commutative 28.4%

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

   10. Simplified 28.4%

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

   11. Taylor expanded in x around 0 26.7%

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

       1. associate--l+ 26.7%

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

   13. Simplified 26.7%

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

   if 7e7 < y

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-+r- 50.1%

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

      4. associate-+l- 22.9%

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

      5. associate-+r- 8.5%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 20.7%

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

      2. +-commutative 20.7%

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

      3. +-commutative 20.7%

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

   7. Simplified 20.7%

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

   8. Taylor expanded in z around inf 18.4%

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

      1. associate--l+ 18.4%

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

      2. +-commutative 18.4%

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

   10. Simplified 18.4%

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

   11. Taylor expanded in y around inf 17.4%

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

       1. distribute-lft-out 17.4%

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

   13. Simplified 17.4%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 70000000:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 550000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 550000.0)
   (- (+ (sqrt (+ z 1.0)) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
   (+
    (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x)))
    1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 550000.0) {
		tmp = (sqrt((z + 1.0)) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	} else {
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 550000.0d0) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - ((sqrt(y) + sqrt(z)) + sqrt(x))
    else
        tmp = ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 550000.0) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - ((Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x));
	} else {
		tmp = ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 550000.0:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - ((math.sqrt(y) + math.sqrt(z)) + math.sqrt(x))
	else:
		tmp = ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 550000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 550000.0)
		tmp = (sqrt((z + 1.0)) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	else
		tmp = ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 550000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.5e5

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+l- 63.5%

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

      6. +-commutative 63.5%

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

      7. +-commutative 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in y around 0 29.3%

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

   6. Taylor expanded in x around 0 14.3%

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

   7. Taylor expanded in t around inf 12.8%

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

   if 5.5e5 < 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. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-+r- 70.1%

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

      4. associate-+l- 51.7%

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

      5. associate-+r- 51.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.5%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in z around inf 27.0%

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

      1. associate--l+ 27.0%

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

      2. +-commutative 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in x around 0 13.8%

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

       1. associate--l+ 32.5%

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

   13. Simplified 32.5%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 550000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 25000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 25000000000000.0)
   (- (+ (sqrt (+ z 1.0)) 2.0) (+ (sqrt z) (sqrt x)))
   (+ (sqrt (+ x 1.0)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25000000000000.0) {
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(x));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 25000000000000.0d0) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - (sqrt(z) + sqrt(x))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25000000000000.0) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - (Math.sqrt(z) + Math.sqrt(x));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 25000000000000.0:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - (math.sqrt(z) + math.sqrt(x))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 25000000000000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(sqrt(z) + sqrt(x)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(y + 1.0)) - 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)
	tmp = 0.0;
	if (z <= 25000000000000.0)
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(x));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 25000000000000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e13

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-+r- 74.0%

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

      4. associate-+l- 62.6%

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

      5. associate-+r- 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around inf 16.6%

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

      1. associate--l+ 21.5%

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

      2. +-commutative 21.5%

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

      3. +-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in z around inf 17.2%

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

   9. Taylor expanded in x around 0 14.6%

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

   10. Taylor expanded in y around 0 26.8%

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

   if 2.5e13 < z

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. associate-+r+ 87.4%

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

      3. associate-+r- 69.9%

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

      4. associate-+l- 52.1%

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

      5. associate-+r- 52.1%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 21.1%

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

      2. +-commutative 21.1%

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

      3. +-commutative 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in z around inf 27.5%

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

      1. +-commutative 27.5%

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

   10. Simplified 27.5%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 25000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 25000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 25000000000000.0)
   (- (+ (sqrt (+ z 1.0)) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
   (+ (sqrt (+ x 1.0)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25000000000000.0) {
		tmp = (sqrt((z + 1.0)) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 25000000000000.0d0) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - ((sqrt(y) + sqrt(z)) + sqrt(x))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25000000000000.0) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - ((Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 25000000000000.0:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - ((math.sqrt(y) + math.sqrt(z)) + math.sqrt(x))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 25000000000000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(y + 1.0)) - 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)
	tmp = 0.0;
	if (z <= 25000000000000.0)
		tmp = (sqrt((z + 1.0)) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 25000000000000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e13

   1. Initial program 96.8%

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

      1. associate-+l+ 96.8%

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

      2. associate-+l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. +-commutative 96.8%

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

      5. associate-+l- 63.4%

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

      6. +-commutative 63.4%

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

      7. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in y around 0 29.0%

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

   6. Taylor expanded in x around 0 14.0%

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

   7. Taylor expanded in t around inf 12.5%

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

   if 2.5e13 < z

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. associate-+r+ 87.4%

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

      3. associate-+r- 69.9%

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

      4. associate-+l- 52.1%

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

      5. associate-+r- 52.1%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 21.1%

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

      2. +-commutative 21.1%

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

      3. +-commutative 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in z around inf 27.5%

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

      1. +-commutative 27.5%

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

   10. Simplified 27.5%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 25000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.96:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.96)
   (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt z) (sqrt x))))
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.96) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(x)));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.96d0) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(z) + sqrt(x)))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.96) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + Math.sqrt(x)));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.96:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(z) + math.sqrt(x)))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.96)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + sqrt(x))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.96)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(x)));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.96], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.96

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.4%

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

      5. associate-+r- 97.4%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in t around inf 17.7%

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

      1. associate--l+ 21.8%

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

      2. +-commutative 21.8%

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

      3. +-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in z around inf 21.8%

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

   9. Taylor expanded in x around 0 25.0%

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

   10. Taylor expanded in y around 0 14.9%

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

       1. associate--l+ 27.6%

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

   12. Simplified 27.6%

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

   if 1.96 < y

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-+r- 50.9%

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

      4. associate-+l- 24.2%

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

      5. associate-+r- 10.2%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in x around inf 20.0%

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

      1. neg-mul-1 20.0%

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

   10. Simplified 20.0%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.96:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.72:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.72)
   (+ (- (sqrt (+ y 1.0)) (sqrt x)) 1.0)
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.72) {
		tmp = (sqrt((y + 1.0)) - sqrt(x)) + 1.0;
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.72d0) then
        tmp = (sqrt((y + 1.0d0)) - sqrt(x)) + 1.0d0
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.72) {
		tmp = (Math.sqrt((y + 1.0)) - Math.sqrt(x)) + 1.0;
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.72:
		tmp = (math.sqrt((y + 1.0)) - math.sqrt(x)) + 1.0
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.72)
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(x)) + 1.0);
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.72)
		tmp = (sqrt((y + 1.0)) - sqrt(x)) + 1.0;
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.72], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.71999999999999997

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.4%

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

      5. associate-+r- 97.4%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in t around inf 17.7%

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

      1. associate--l+ 21.8%

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

      2. +-commutative 21.8%

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

      3. +-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in z around inf 21.8%

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

   9. Taylor expanded in x around 0 25.0%

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

   10. Taylor expanded in z around inf 16.4%

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

       1. associate--l+ 16.4%

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

   12. Simplified 16.4%

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

   if 0.71999999999999997 < y

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-+r- 50.9%

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

      4. associate-+l- 24.2%

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

      5. associate-+r- 10.2%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in x around inf 20.0%

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

      1. neg-mul-1 20.0%

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

   10. Simplified 20.0%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.72:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;\left(\sqrt{y + 1} + 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.5], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.4%

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

      4. associate-+l- 97.4%

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

      5. associate-+r- 97.4%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in t around inf 17.7%

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

      1. associate--l+ 21.8%

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

      2. +-commutative 21.8%

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

      3. +-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in z around inf 21.8%

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

   9. Taylor expanded in x around 0 25.0%

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

   10. Taylor expanded in z around inf 16.4%

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

   if 1.5 < y

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-+r- 50.9%

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

      4. associate-+l- 24.2%

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

      5. associate-+r- 10.2%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 20.8%

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

      2. +-commutative 20.8%

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

      3. +-commutative 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in x around inf 20.0%

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

      1. neg-mul-1 20.0%

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

   10. Simplified 20.0%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;\left(\sqrt{y + 1} + 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. associate-+r+ 92.2%

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

   3. associate-+r- 72.0%

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

   4. associate-+l- 57.4%

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

   5. associate-+r- 49.7%

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

3. Simplified 39.5%

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

5. Taylor expanded in t around inf 10.1%

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

   1. associate--l+ 21.3%

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

   2. +-commutative 21.3%

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

   3. +-commutative 21.3%

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

7. Simplified 21.3%

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

8. Taylor expanded in x around inf 15.9%

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

   1. neg-mul-1 15.9%

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

10. Simplified 15.9%

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

11. Final simplification 15.9%

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

Alternative 18: 15.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. associate-+r+ 92.2%

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

   3. associate-+r- 72.0%

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

   4. associate-+l- 57.4%

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

   5. associate-+r- 49.7%

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

3. Simplified 39.5%

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

5. Taylor expanded in t around inf 10.1%

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

   1. associate--l+ 21.3%

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

   2. +-commutative 21.3%

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

   3. +-commutative 21.3%

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

7. Simplified 21.3%

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

8. Taylor expanded in z around inf 14.4%

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

9. Taylor expanded in x around 0 18.3%

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

10. Taylor expanded in y around inf 16.8%

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

11. Final simplification 16.8%

    \[\leadsto \sqrt{y} + 1 \]
12. Add Preprocessing

Alternative 19: 33.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. associate-+r+ 92.2%

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

   3. associate-+r- 72.0%

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

   4. associate-+l- 57.4%

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

   5. associate-+r- 49.7%

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

3. Simplified 39.5%

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

5. Taylor expanded in t around inf 10.1%

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

   1. associate--l+ 21.3%

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

   2. +-commutative 21.3%

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

   3. +-commutative 21.3%

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

7. Simplified 21.3%

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

8. Taylor expanded in z around inf 14.4%

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

9. Taylor expanded in x around 0 18.3%

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

10. Taylor expanded in x around inf 14.6%

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

    1. mul-1-neg 14.6%

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

12. Simplified 14.6%

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

13. Final simplification 14.6%

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

Alternative 20: 7.6% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 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[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. associate-+r+ 92.2%

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

   3. associate-+r- 72.0%

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

   4. associate-+l- 57.4%

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

   5. associate-+r- 49.7%

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

3. Simplified 39.5%

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

5. Taylor expanded in t around inf 10.1%

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

   1. associate--l+ 21.3%

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

   2. +-commutative 21.3%

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

   3. +-commutative 21.3%

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

7. Simplified 21.3%

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

8. Taylor expanded in z around inf 14.4%

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

9. Taylor expanded in x around 0 18.3%

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

10. Taylor expanded in y around inf 6.9%

    \[\leadsto \color{blue}{\sqrt{y}} \]

11. Final simplification 6.9%

    \[\leadsto \sqrt{y} \]
12. Add Preprocessing

Alternative 21: 3.1% accurate, 274.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x * 0.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. associate-+r+ 92.2%

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

   3. associate-+r- 72.0%

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

   4. associate-+l- 57.4%

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

   5. associate-+r- 49.7%

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

3. Simplified 39.5%

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

5. Taylor expanded in t around inf 10.1%

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

   1. associate--l+ 21.3%

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

   2. +-commutative 21.3%

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

   3. +-commutative 21.3%

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

7. Simplified 21.3%

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

8. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

10. Simplified 21.2%

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

11. Taylor expanded in x around inf 3.1%

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

    1. distribute-rgt1-in 3.1%

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

    2. metadata-eval 3.1%

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

    3. mul0-lft 3.1%

       \[\leadsto x \cdot \color{blue}{0} \]

13. Simplified 3.1%

    \[\leadsto \color{blue}{x \cdot 0} \]

14. Final simplification 3.1%

    \[\leadsto x \cdot 0 \]
15. Add Preprocessing

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024115 
(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))))