Main:z from

Percentage Accurate: 91.3% → 99.8%

Time: 35.5s

Alternatives: 22

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 + \sqrt{x}\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 + \sqrt{t}\\ t_5 := \sqrt{y + 1}\\ t_6 := t\_5 + \sqrt{y}\\ t_7 := t\_5 - \sqrt{y}\\ t_8 := \sqrt{z} + \sqrt{z + 1}\\ \mathbf{if}\;t\_7 \leq 0.5:\\ \;\;\;\;\frac{t\_6 + t\_2}{t\_6 \cdot t\_2} + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(t\_3 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + \left(t\_7 + \frac{t\_4 + t\_8}{t\_4 \cdot t\_8}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (+ t_1 (sqrt x)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (+ t_3 (sqrt t)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ t_5 (sqrt y)))
        (t_7 (- t_5 (sqrt y)))
        (t_8 (+ (sqrt z) (sqrt (+ z 1.0)))))
   (if (<= t_7 0.5)
     (+
      (/ (+ t_6 t_2) (* t_6 t_2))
      (+ (* 0.5 (sqrt (/ 1.0 z))) (- t_3 (sqrt t))))
     (+ (- t_1 (sqrt x)) (+ t_7 (/ (+ t_4 t_8) (* t_4 t_8)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = t_1 + sqrt(x);
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 + sqrt(t);
	double t_5 = sqrt((y + 1.0));
	double t_6 = t_5 + sqrt(y);
	double t_7 = t_5 - sqrt(y);
	double t_8 = sqrt(z) + sqrt((z + 1.0));
	double tmp;
	if (t_7 <= 0.5) {
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * sqrt((1.0 / z))) + (t_3 - sqrt(t)));
	} else {
		tmp = (t_1 - sqrt(x)) + (t_7 + ((t_4 + t_8) / (t_4 * t_8)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = t_1 + sqrt(x)
    t_3 = sqrt((t + 1.0d0))
    t_4 = t_3 + sqrt(t)
    t_5 = sqrt((y + 1.0d0))
    t_6 = t_5 + sqrt(y)
    t_7 = t_5 - sqrt(y)
    t_8 = sqrt(z) + sqrt((z + 1.0d0))
    if (t_7 <= 0.5d0) then
        tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5d0 * sqrt((1.0d0 / z))) + (t_3 - sqrt(t)))
    else
        tmp = (t_1 - sqrt(x)) + (t_7 + ((t_4 + t_8) / (t_4 * t_8)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = t_1 + Math.sqrt(x);
	double t_3 = Math.sqrt((t + 1.0));
	double t_4 = t_3 + Math.sqrt(t);
	double t_5 = Math.sqrt((y + 1.0));
	double t_6 = t_5 + Math.sqrt(y);
	double t_7 = t_5 - Math.sqrt(y);
	double t_8 = Math.sqrt(z) + Math.sqrt((z + 1.0));
	double tmp;
	if (t_7 <= 0.5) {
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * Math.sqrt((1.0 / z))) + (t_3 - Math.sqrt(t)));
	} else {
		tmp = (t_1 - Math.sqrt(x)) + (t_7 + ((t_4 + t_8) / (t_4 * t_8)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = t_1 + math.sqrt(x)
	t_3 = math.sqrt((t + 1.0))
	t_4 = t_3 + math.sqrt(t)
	t_5 = math.sqrt((y + 1.0))
	t_6 = t_5 + math.sqrt(y)
	t_7 = t_5 - math.sqrt(y)
	t_8 = math.sqrt(z) + math.sqrt((z + 1.0))
	tmp = 0
	if t_7 <= 0.5:
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * math.sqrt((1.0 / z))) + (t_3 - math.sqrt(t)))
	else:
		tmp = (t_1 - math.sqrt(x)) + (t_7 + ((t_4 + t_8) / (t_4 * t_8)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(t_1 + sqrt(x))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 + sqrt(t))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(t_5 + sqrt(y))
	t_7 = Float64(t_5 - sqrt(y))
	t_8 = Float64(sqrt(z) + sqrt(Float64(z + 1.0)))
	tmp = 0.0
	if (t_7 <= 0.5)
		tmp = Float64(Float64(Float64(t_6 + t_2) / Float64(t_6 * t_2)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(t_3 - sqrt(t))));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(t_7 + Float64(Float64(t_4 + t_8) / Float64(t_4 * t_8))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = t_1 + sqrt(x);
	t_3 = sqrt((t + 1.0));
	t_4 = t_3 + sqrt(t);
	t_5 = sqrt((y + 1.0));
	t_6 = t_5 + sqrt(y);
	t_7 = t_5 - sqrt(y);
	t_8 = sqrt(z) + sqrt((z + 1.0));
	tmp = 0.0;
	if (t_7 <= 0.5)
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * sqrt((1.0 / z))) + (t_3 - sqrt(t)));
	else
		tmp = (t_1 - sqrt(x)) + (t_7 + ((t_4 + t_8) / (t_4 * t_8)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.5], N[(N[(N[(t$95$6 + t$95$2), $MachinePrecision] / N[(t$95$6 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$7 + N[(N[(t$95$4 + t$95$8), $MachinePrecision] / N[(t$95$4 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.5

   1. Initial program 82.0%

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

      1. associate-+l+ 82.0%

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

      2. associate-+l- 81.4%

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

      3. associate-+l- 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

      6. +-commutative 82.0%

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

   3. Simplified 82.0%

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

      1. +-commutative 82.0%

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

      2. flip-- 82.8%

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

      3. flip-- 82.8%

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

      4. frac-add 82.8%

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

   6. Applied egg-rr 83.6%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in z around inf 59.0%

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

   if 0.5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. associate-+l+ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+l- 78.8%

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

      6. +-commutative 78.8%

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

      7. +-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. associate--r- 98.2%

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

      2. flip-- 98.3%

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

      3. flip-- 98.3%

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

      4. frac-add 98.3%

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

   6. Applied egg-rr 98.7%

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

   8. Simplified 99.4%

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

4. Final simplification 80.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := t\_3 + \sqrt{x}\\ t_5 := \sqrt{y + 1}\\ t_6 := \left(t\_3 - \sqrt{x}\right) + \left(t\_5 - \sqrt{y}\right)\\ t_7 := t\_5 + \sqrt{y}\\ \mathbf{if}\;t\_6 + t\_1 \leq 2.0005:\\ \;\;\;\;\frac{t\_7 + t\_4}{t\_7 \cdot t\_4} + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(t\_2 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6 + \left(t\_1 + \frac{1}{t\_2 + \sqrt{t}}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ t_3 (sqrt x)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ (- t_3 (sqrt x)) (- t_5 (sqrt y))))
        (t_7 (+ t_5 (sqrt y))))
   (if (<= (+ t_6 t_1) 2.0005)
     (+
      (/ (+ t_7 t_4) (* t_7 t_4))
      (+ (* 0.5 (sqrt (/ 1.0 z))) (- t_2 (sqrt t))))
     (+ t_6 (+ t_1 (/ 1.0 (+ t_2 (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_3 + sqrt(x);
	double t_5 = sqrt((y + 1.0));
	double t_6 = (t_3 - sqrt(x)) + (t_5 - sqrt(y));
	double t_7 = t_5 + sqrt(y);
	double tmp;
	if ((t_6 + t_1) <= 2.0005) {
		tmp = ((t_7 + t_4) / (t_7 * t_4)) + ((0.5 * sqrt((1.0 / z))) + (t_2 - sqrt(t)));
	} else {
		tmp = t_6 + (t_1 + (1.0 / (t_2 + sqrt(t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0))
    t_3 = sqrt((x + 1.0d0))
    t_4 = t_3 + sqrt(x)
    t_5 = sqrt((y + 1.0d0))
    t_6 = (t_3 - sqrt(x)) + (t_5 - sqrt(y))
    t_7 = t_5 + sqrt(y)
    if ((t_6 + t_1) <= 2.0005d0) then
        tmp = ((t_7 + t_4) / (t_7 * t_4)) + ((0.5d0 * sqrt((1.0d0 / z))) + (t_2 - sqrt(t)))
    else
        tmp = t_6 + (t_1 + (1.0d0 / (t_2 + sqrt(t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0));
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = t_3 + Math.sqrt(x);
	double t_5 = Math.sqrt((y + 1.0));
	double t_6 = (t_3 - Math.sqrt(x)) + (t_5 - Math.sqrt(y));
	double t_7 = t_5 + Math.sqrt(y);
	double tmp;
	if ((t_6 + t_1) <= 2.0005) {
		tmp = ((t_7 + t_4) / (t_7 * t_4)) + ((0.5 * Math.sqrt((1.0 / z))) + (t_2 - Math.sqrt(t)));
	} else {
		tmp = t_6 + (t_1 + (1.0 / (t_2 + Math.sqrt(t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0))
	t_3 = math.sqrt((x + 1.0))
	t_4 = t_3 + math.sqrt(x)
	t_5 = math.sqrt((y + 1.0))
	t_6 = (t_3 - math.sqrt(x)) + (t_5 - math.sqrt(y))
	t_7 = t_5 + math.sqrt(y)
	tmp = 0
	if (t_6 + t_1) <= 2.0005:
		tmp = ((t_7 + t_4) / (t_7 * t_4)) + ((0.5 * math.sqrt((1.0 / z))) + (t_2 - math.sqrt(t)))
	else:
		tmp = t_6 + (t_1 + (1.0 / (t_2 + math.sqrt(t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_3 + sqrt(x))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(Float64(t_3 - sqrt(x)) + Float64(t_5 - sqrt(y)))
	t_7 = Float64(t_5 + sqrt(y))
	tmp = 0.0
	if (Float64(t_6 + t_1) <= 2.0005)
		tmp = Float64(Float64(Float64(t_7 + t_4) / Float64(t_7 * t_4)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(t_2 - sqrt(t))));
	else
		tmp = Float64(t_6 + Float64(t_1 + Float64(1.0 / Float64(t_2 + sqrt(t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0));
	t_3 = sqrt((x + 1.0));
	t_4 = t_3 + sqrt(x);
	t_5 = sqrt((y + 1.0));
	t_6 = (t_3 - sqrt(x)) + (t_5 - sqrt(y));
	t_7 = t_5 + sqrt(y);
	tmp = 0.0;
	if ((t_6 + t_1) <= 2.0005)
		tmp = ((t_7 + t_4) / (t_7 * t_4)) + ((0.5 * sqrt((1.0 / z))) + (t_2 - sqrt(t)));
	else
		tmp = t_6 + (t_1 + (1.0 / (t_2 + sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$6 + t$95$1), $MachinePrecision], 2.0005], N[(N[(N[(t$95$7 + t$95$4), $MachinePrecision] / N[(t$95$7 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$6 + N[(t$95$1 + N[(1.0 / N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00050000000000017

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-+l- 65.5%

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

      3. associate-+l- 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. +-commutative 89.3%

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

   3. Simplified 89.3%

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

      1. +-commutative 89.3%

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

      2. flip-- 89.7%

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

      3. flip-- 89.8%

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

      4. frac-add 89.8%

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

   6. Applied egg-rr 90.2%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in z around inf 62.0%

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

   if 2.00050000000000017 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.7%

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

      1. associate-+l+ 98.7%

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

      2. associate-+l- 98.7%

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

      3. associate-+l- 98.7%

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

      4. +-commutative 98.7%

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

      5. +-commutative 98.7%

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

      6. +-commutative 98.7%

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

   3. Simplified 98.7%

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

      1. flip-- 98.6%

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

      2. add-sqr-sqrt 77.9%

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

      3. add-sqr-sqrt 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-inverses 99.2%

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

      3. metadata-eval 99.2%

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

      4. +-commutative 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.0005:\\ \;\;\;\;\frac{\left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 + \sqrt{x}\\ t_3 := \sqrt{z} + \sqrt{z + 1}\\ t_4 := \sqrt{t + 1}\\ t_5 := \sqrt{y + 1}\\ t_6 := t\_5 + \sqrt{y}\\ t_7 := t\_4 + \sqrt{t}\\ \mathbf{if}\;t\_5 - \sqrt{y} \leq 0.9995:\\ \;\;\;\;\frac{t\_6 + t\_2}{t\_6 \cdot t\_2} + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(t\_4 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + \left(\frac{t\_7 + t\_3}{t\_7 \cdot t\_3} + \left(\left(y \cdot 0.5 - \sqrt{y}\right) + 1\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (+ t_1 (sqrt x)))
        (t_3 (+ (sqrt z) (sqrt (+ z 1.0))))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ t_5 (sqrt y)))
        (t_7 (+ t_4 (sqrt t))))
   (if (<= (- t_5 (sqrt y)) 0.9995)
     (+
      (/ (+ t_6 t_2) (* t_6 t_2))
      (+ (* 0.5 (sqrt (/ 1.0 z))) (- t_4 (sqrt t))))
     (+
      (- t_1 (sqrt x))
      (+ (/ (+ t_7 t_3) (* t_7 t_3)) (+ (- (* y 0.5) (sqrt y)) 1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = t_1 + sqrt(x);
	double t_3 = sqrt(z) + sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0));
	double t_5 = sqrt((y + 1.0));
	double t_6 = t_5 + sqrt(y);
	double t_7 = t_4 + sqrt(t);
	double tmp;
	if ((t_5 - sqrt(y)) <= 0.9995) {
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * sqrt((1.0 / z))) + (t_4 - sqrt(t)));
	} else {
		tmp = (t_1 - sqrt(x)) + (((t_7 + t_3) / (t_7 * t_3)) + (((y * 0.5) - sqrt(y)) + 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = t_1 + sqrt(x)
    t_3 = sqrt(z) + sqrt((z + 1.0d0))
    t_4 = sqrt((t + 1.0d0))
    t_5 = sqrt((y + 1.0d0))
    t_6 = t_5 + sqrt(y)
    t_7 = t_4 + sqrt(t)
    if ((t_5 - sqrt(y)) <= 0.9995d0) then
        tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5d0 * sqrt((1.0d0 / z))) + (t_4 - sqrt(t)))
    else
        tmp = (t_1 - sqrt(x)) + (((t_7 + t_3) / (t_7 * t_3)) + (((y * 0.5d0) - sqrt(y)) + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = t_1 + Math.sqrt(x);
	double t_3 = Math.sqrt(z) + Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt((t + 1.0));
	double t_5 = Math.sqrt((y + 1.0));
	double t_6 = t_5 + Math.sqrt(y);
	double t_7 = t_4 + Math.sqrt(t);
	double tmp;
	if ((t_5 - Math.sqrt(y)) <= 0.9995) {
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * Math.sqrt((1.0 / z))) + (t_4 - Math.sqrt(t)));
	} else {
		tmp = (t_1 - Math.sqrt(x)) + (((t_7 + t_3) / (t_7 * t_3)) + (((y * 0.5) - Math.sqrt(y)) + 1.0));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = t_1 + math.sqrt(x)
	t_3 = math.sqrt(z) + math.sqrt((z + 1.0))
	t_4 = math.sqrt((t + 1.0))
	t_5 = math.sqrt((y + 1.0))
	t_6 = t_5 + math.sqrt(y)
	t_7 = t_4 + math.sqrt(t)
	tmp = 0
	if (t_5 - math.sqrt(y)) <= 0.9995:
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * math.sqrt((1.0 / z))) + (t_4 - math.sqrt(t)))
	else:
		tmp = (t_1 - math.sqrt(x)) + (((t_7 + t_3) / (t_7 * t_3)) + (((y * 0.5) - math.sqrt(y)) + 1.0))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(t_1 + sqrt(x))
	t_3 = Float64(sqrt(z) + sqrt(Float64(z + 1.0)))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(t_5 + sqrt(y))
	t_7 = Float64(t_4 + sqrt(t))
	tmp = 0.0
	if (Float64(t_5 - sqrt(y)) <= 0.9995)
		tmp = Float64(Float64(Float64(t_6 + t_2) / Float64(t_6 * t_2)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(t_4 - sqrt(t))));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(Float64(t_7 + t_3) / Float64(t_7 * t_3)) + Float64(Float64(Float64(y * 0.5) - sqrt(y)) + 1.0)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = t_1 + sqrt(x);
	t_3 = sqrt(z) + sqrt((z + 1.0));
	t_4 = sqrt((t + 1.0));
	t_5 = sqrt((y + 1.0));
	t_6 = t_5 + sqrt(y);
	t_7 = t_4 + sqrt(t);
	tmp = 0.0;
	if ((t_5 - sqrt(y)) <= 0.9995)
		tmp = ((t_6 + t_2) / (t_6 * t_2)) + ((0.5 * sqrt((1.0 / z))) + (t_4 - sqrt(t)));
	else
		tmp = (t_1 - sqrt(x)) + (((t_7 + t_3) / (t_7 * t_3)) + (((y * 0.5) - sqrt(y)) + 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.9995], N[(N[(N[(t$95$6 + t$95$2), $MachinePrecision] / N[(t$95$6 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$7 + t$95$3), $MachinePrecision] / N[(t$95$7 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.99950000000000006

   1. Initial program 82.0%

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

      1. associate-+l+ 82.0%

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

      2. associate-+l- 81.4%

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

      3. associate-+l- 82.0%

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

      4. +-commutative 82.0%

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

      5. +-commutative 82.0%

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

      6. +-commutative 82.0%

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

   3. Simplified 82.0%

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

      1. +-commutative 82.0%

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

      2. flip-- 82.8%

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

      3. flip-- 82.8%

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

      4. frac-add 82.8%

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

   6. Applied egg-rr 83.6%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in z around inf 59.0%

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

   if 0.99950000000000006 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. associate-+l+ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+l- 78.8%

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

      6. +-commutative 78.8%

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

      7. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in y around 0 78.8%

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

      1. associate--l+ 78.8%

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

   7. Simplified 78.8%

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

      1. associate--r- 98.2%

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

      2. flip-- 98.3%

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

      3. flip-- 98.3%

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

      4. frac-add 98.3%

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

   9. Applied egg-rr 98.7%

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

   11. Simplified 99.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0.9995:\\ \;\;\;\;\frac{\left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{\left(\sqrt{t + 1} + \sqrt{t}\right) + \left(\sqrt{z} + \sqrt{z + 1}\right)}{\left(\sqrt{t + 1} + \sqrt{t}\right) \cdot \left(\sqrt{z} + \sqrt{z + 1}\right)} + \left(\left(y \cdot 0.5 - \sqrt{y}\right) + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (sqrt (+ z 1.0))))
   (if (<= t_2 2e-5)
     (+
      (/ 1.0 (+ t_1 (sqrt x)))
      (+ (* 0.5 (sqrt (/ 1.0 y))) (+ t_3 (- (- t_4 (sqrt z)) (sqrt t)))))
     (+
      (+ (- t_1 (sqrt x)) t_2)
      (+ (- t_3 (sqrt t)) (/ 1.0 (+ (sqrt z) t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double t_3 = sqrt((t + 1.0));
	double t_4 = sqrt((z + 1.0));
	double tmp;
	if (t_2 <= 2e-5) {
		tmp = (1.0 / (t_1 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_3 + ((t_4 - sqrt(z)) - sqrt(t))));
	} else {
		tmp = ((t_1 - sqrt(x)) + t_2) + ((t_3 - sqrt(t)) + (1.0 / (sqrt(z) + t_4)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    t_3 = sqrt((t + 1.0d0))
    t_4 = sqrt((z + 1.0d0))
    if (t_2 <= 2d-5) then
        tmp = (1.0d0 / (t_1 + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) + (t_3 + ((t_4 - sqrt(z)) - sqrt(t))))
    else
        tmp = ((t_1 - sqrt(x)) + t_2) + ((t_3 - sqrt(t)) + (1.0d0 / (sqrt(z) + t_4)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_3 = Math.sqrt((t + 1.0));
	double t_4 = Math.sqrt((z + 1.0));
	double tmp;
	if (t_2 <= 2e-5) {
		tmp = (1.0 / (t_1 + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) + (t_3 + ((t_4 - Math.sqrt(z)) - Math.sqrt(t))));
	} else {
		tmp = ((t_1 - Math.sqrt(x)) + t_2) + ((t_3 - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + t_4)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_3 = math.sqrt((t + 1.0))
	t_4 = math.sqrt((z + 1.0))
	tmp = 0
	if t_2 <= 2e-5:
		tmp = (1.0 / (t_1 + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) + (t_3 + ((t_4 - math.sqrt(z)) - math.sqrt(t))))
	else:
		tmp = ((t_1 - math.sqrt(x)) + t_2) + ((t_3 - math.sqrt(t)) + (1.0 / (math.sqrt(z) + t_4)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t_2 <= 2e-5)
		tmp = Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(t_3 + Float64(Float64(t_4 - sqrt(z)) - sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) + t_2) + Float64(Float64(t_3 - sqrt(t)) + Float64(1.0 / Float64(sqrt(z) + t_4))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	t_3 = sqrt((t + 1.0));
	t_4 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t_2 <= 2e-5)
		tmp = (1.0 / (t_1 + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_3 + ((t_4 - sqrt(z)) - sqrt(t))));
	else
		tmp = ((t_1 - sqrt(x)) + t_2) + ((t_3 - sqrt(t)) + (1.0 / (sqrt(z) + t_4)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 2.00000000000000016e-5

   1. Initial program 81.8%

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

      1. associate-+l+ 81.8%

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

      2. associate-+l+ 81.8%

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

      3. +-commutative 81.8%

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

      4. +-commutative 81.8%

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

      5. associate-+l- 70.5%

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

      6. +-commutative 70.5%

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

      7. +-commutative 70.5%

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

   3. Simplified 70.5%

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

      1. flip-- 70.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. +-commutative 55.5%

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

      4. add-sqr-sqrt 70.5%

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

      5. +-commutative 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. associate--l+ 72.4%

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

      2. +-inverses 72.4%

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

      3. metadata-eval 72.4%

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

      4. +-commutative 72.4%

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

   8. Simplified 72.4%

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

   9. Taylor expanded in y around inf 76.8%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. associate-+l- 61.8%

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

      3. associate-+l- 98.0%

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

      4. +-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   3. Simplified 98.0%

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

      1. flip-- 98.0%

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

      2. add-sqr-sqrt 81.3%

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

      3. add-sqr-sqrt 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. associate--l+ 98.8%

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

      2. +-inverses 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 88.9%

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

Alternative 5: 98.9% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ t 1.0))) (t_3 (sqrt (+ z 1.0))))
   (if (<= (- t_1 (sqrt y)) 2e-5)
     (+
      (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
      (+ (* 0.5 (sqrt (/ 1.0 y))) (+ t_2 (- (- t_3 (sqrt z)) (sqrt t)))))
     (+
      (+ (- t_2 (sqrt t)) (/ 1.0 (+ (sqrt z) t_3)))
      (- (+ t_1 1.0) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((t + 1.0));
	double t_3 = sqrt((z + 1.0));
	double tmp;
	if ((t_1 - sqrt(y)) <= 2e-5) {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))));
	} else {
		tmp = ((t_2 - sqrt(t)) + (1.0 / (sqrt(z) + t_3))) + ((t_1 + 1.0) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((t + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    if ((t_1 - sqrt(y)) <= 2d-5) then
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + ((0.5d0 * sqrt((1.0d0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))))
    else
        tmp = ((t_2 - sqrt(t)) + (1.0d0 / (sqrt(z) + t_3))) + ((t_1 + 1.0d0) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((t + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(y)) <= 2e-5) {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + ((0.5 * Math.sqrt((1.0 / y))) + (t_2 + ((t_3 - Math.sqrt(z)) - Math.sqrt(t))));
	} else {
		tmp = ((t_2 - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + t_3))) + ((t_1 + 1.0) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((t + 1.0))
	t_3 = math.sqrt((z + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(y)) <= 2e-5:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + ((0.5 * math.sqrt((1.0 / y))) + (t_2 + ((t_3 - math.sqrt(z)) - math.sqrt(t))))
	else:
		tmp = ((t_2 - math.sqrt(t)) + (1.0 / (math.sqrt(z) + t_3))) + ((t_1 + 1.0) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(y)) <= 2e-5)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(t_2 + Float64(Float64(t_3 - sqrt(z)) - sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(t_2 - sqrt(t)) + Float64(1.0 / Float64(sqrt(z) + t_3))) + Float64(Float64(t_1 + 1.0) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((t + 1.0));
	t_3 = sqrt((z + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(y)) <= 2e-5)
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) + ((0.5 * sqrt((1.0 / y))) + (t_2 + ((t_3 - sqrt(z)) - sqrt(t))));
	else
		tmp = ((t_2 - sqrt(t)) + (1.0 / (sqrt(z) + t_3))) + ((t_1 + 1.0) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 2.00000000000000016e-5

   1. Initial program 81.8%

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

      1. associate-+l+ 81.8%

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

      2. associate-+l+ 81.8%

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

      3. +-commutative 81.8%

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

      4. +-commutative 81.8%

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

      5. associate-+l- 70.5%

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

      6. +-commutative 70.5%

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

      7. +-commutative 70.5%

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

   3. Simplified 70.5%

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

      1. flip-- 70.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. +-commutative 55.5%

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

      4. add-sqr-sqrt 70.5%

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

      5. +-commutative 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. associate--l+ 72.4%

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

      2. +-inverses 72.4%

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

      3. metadata-eval 72.4%

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

      4. +-commutative 72.4%

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

   8. Simplified 72.4%

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

   9. Taylor expanded in y around inf 76.8%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. associate-+l- 61.8%

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

      3. associate-+l- 98.0%

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

      4. +-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   3. Simplified 98.0%

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

      1. flip-- 98.0%

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

      2. add-sqr-sqrt 81.3%

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

      3. add-sqr-sqrt 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. associate--l+ 98.8%

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

      2. +-inverses 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in x around 0 49.0%

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

       1. +-commutative 49.0%

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

   11. Simplified 49.0%

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

4. Final simplification 61.5%

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

Alternative 6: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \frac{1}{\sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_1 5e-8)
     (+
      (+ (* 0.5 (sqrt (/ 1.0 z))) (- (sqrt (+ t 1.0)) (sqrt t)))
      (+ (* 0.5 (sqrt (/ 1.0 y))) (* 0.5 (/ 1.0 (sqrt x)))))
     (+ t_1 (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- (sqrt (+ z 1.0)) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_1 <= 5e-8) {
		tmp = ((0.5 * sqrt((1.0 / z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((0.5 * sqrt((1.0 / y))) + (0.5 * (1.0 / sqrt(x))));
	} else {
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (sqrt((z + 1.0)) - sqrt(z)));
	}
	return tmp;
}

Fortran

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

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_1 <= 5e-8) {
		tmp = ((0.5 * Math.sqrt((1.0 / z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) + ((0.5 * Math.sqrt((1.0 / y))) + (0.5 * (1.0 / Math.sqrt(x))));
	} else {
		tmp = t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_1 <= 5e-8:
		tmp = ((0.5 * math.sqrt((1.0 / z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) + ((0.5 * math.sqrt((1.0 / y))) + (0.5 * (1.0 / math.sqrt(x))))
	else:
		tmp = t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (math.sqrt((z + 1.0)) - math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(0.5 * Float64(1.0 / sqrt(x)))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_1 <= 5e-8)
		tmp = ((0.5 * sqrt((1.0 / z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((0.5 * sqrt((1.0 / y))) + (0.5 * (1.0 / sqrt(x))));
	else
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (sqrt((z + 1.0)) - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-8], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 4.9999999999999998e-8

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. associate-+l- 43.2%

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

      3. associate-+l- 83.9%

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

      4. +-commutative 83.9%

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

      5. +-commutative 83.9%

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

      6. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in x around inf 85.6%

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

   6. Taylor expanded in y around inf 40.9%

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

   7. Taylor expanded in z around inf 29.7%

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

      1. sqrt-div 29.7%

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

      2. metadata-eval 29.7%

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

   9. Applied egg-rr 29.7%

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

   if 4.9999999999999998e-8 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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- 80.6%

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

      6. +-commutative 80.6%

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

      7. +-commutative 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in t around inf 60.1%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \frac{1}{\sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(\left(\sqrt{y + 1} + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.6e+17)
   (+
    (+ (- (sqrt (+ t 1.0)) (sqrt t)) (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))
    (- (+ (sqrt (+ y 1.0)) 1.0) (+ (sqrt x) (sqrt y))))
   (/ 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 <= 3.6e+17) {
		tmp = ((sqrt((t + 1.0)) - sqrt(t)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))))) + ((sqrt((y + 1.0)) + 1.0) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.6d+17) then
        tmp = ((sqrt((t + 1.0d0)) - sqrt(t)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))) + ((sqrt((y + 1.0d0)) + 1.0d0) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.6e+17) {
		tmp = ((Math.sqrt((t + 1.0)) - Math.sqrt(t)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))))) + ((Math.sqrt((y + 1.0)) + 1.0) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.6e+17:
		tmp = ((math.sqrt((t + 1.0)) - math.sqrt(t)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))) + ((math.sqrt((y + 1.0)) + 1.0) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.6e+17)
		tmp = Float64(Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))) + Float64(Float64(sqrt(Float64(y + 1.0)) + 1.0) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.6e+17)
		tmp = ((sqrt((t + 1.0)) - sqrt(t)) + (1.0 / (sqrt(z) + sqrt((z + 1.0))))) + ((sqrt((y + 1.0)) + 1.0) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3.6e+17], N[(N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e17

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l- 61.8%

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

      3. associate-+l- 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

      6. +-commutative 96.9%

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

   3. Simplified 96.9%

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

      1. flip-- 96.9%

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

      2. add-sqr-sqrt 79.7%

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

      3. add-sqr-sqrt 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around 0 49.4%

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

       1. +-commutative 49.4%

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

   11. Simplified 49.4%

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

   if 3.6e17 < y

   1. Initial program 82.5%

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

      1. associate-+l+ 82.5%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in t around inf 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+ 24.5%

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

      2. +-commutative 24.5%

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

      3. +-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in x around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cube-cbrt 25.1%

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

       2. pow3 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 24.1%

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

       2. flip-+ 24.0%

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

       3. add-sqr-sqrt 24.4%

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

       4. pow2 24.4%

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

   14. Applied egg-rr 24.4%

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

       1. associate--l+ 26.2%

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

       2. unpow2 26.2%

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

       3. sqr-neg 26.2%

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

       4. rem-square-sqrt 27.1%

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

       5. +-inverses 27.1%

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

       6. metadata-eval 27.1%

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

       7. sub-neg 27.1%

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

       8. remove-double-neg 27.1%

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

       9. +-commutative 27.1%

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

   16. Simplified 27.1%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(\left(\sqrt{y + 1} + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;t\_1 + \left(\left(2 + \left(\sqrt{t + 1} + 0.5 \cdot \left(y + z\right)\right)\right) - \left(\sqrt{z} + \left(\sqrt{t} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t 170000000000.0)
     (+
      t_1
      (-
       (+ 2.0 (+ (sqrt (+ t 1.0)) (* 0.5 (+ y z))))
       (+ (sqrt z) (+ (sqrt t) (sqrt y)))))
     (+
      t_1
      (+
       (- (sqrt (+ y 1.0)) (sqrt y))
       (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + ((2.0 + (sqrt((t + 1.0)) + (0.5 * (y + z)))) - (sqrt(z) + (sqrt(t) + sqrt(y))));
	} else {
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t <= 170000000000.0d0) then
        tmp = t_1 + ((2.0d0 + (sqrt((t + 1.0d0)) + (0.5d0 * (y + z)))) - (sqrt(z) + (sqrt(t) + sqrt(y))))
    else
        tmp = t_1 + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + ((2.0 + (Math.sqrt((t + 1.0)) + (0.5 * (y + z)))) - (Math.sqrt(z) + (Math.sqrt(t) + Math.sqrt(y))));
	} else {
		tmp = t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t <= 170000000000.0:
		tmp = t_1 + ((2.0 + (math.sqrt((t + 1.0)) + (0.5 * (y + z)))) - (math.sqrt(z) + (math.sqrt(t) + math.sqrt(y))))
	else:
		tmp = t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t <= 170000000000.0)
		tmp = Float64(t_1 + Float64(Float64(2.0 + Float64(sqrt(Float64(t + 1.0)) + Float64(0.5 * Float64(y + z)))) - Float64(sqrt(z) + Float64(sqrt(t) + sqrt(y)))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t <= 170000000000.0)
		tmp = t_1 + ((2.0 + (sqrt((t + 1.0)) + (0.5 * (y + z)))) - (sqrt(z) + (sqrt(t) + sqrt(y))));
	else
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 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[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 170000000000.0], N[(t$95$1 + N[(N[(2.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e11

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l+ 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+l- 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around 0 58.3%

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

      1. associate--l+ 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in z around 0 32.3%

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

      1. distribute-lft-out 32.3%

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

      2. associate-+r+ 32.3%

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

      3. +-commutative 32.3%

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

   10. Simplified 32.3%

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

   if 1.7e11 < t

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. +-commutative 82.8%

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

      5. associate-+l- 49.1%

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

      6. +-commutative 49.1%

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

      7. +-commutative 49.1%

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

   3. Simplified 49.1%

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

      1. associate--r- 82.8%

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

      2. flip-- 83.0%

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

      3. flip-- 83.3%

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

      4. frac-add 83.3%

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

   6. Applied egg-rr 83.4%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in t around inf 87.3%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(2 + \left(\sqrt{t + 1} + 0.5 \cdot \left(y + z\right)\right)\right) - \left(\sqrt{z} + \left(\sqrt{t} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;t\_1 + \left(\left(2 + \left(\sqrt{t + 1} + y \cdot 0.5\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t 170000000000.0)
     (+
      t_1
      (-
       (+ 2.0 (+ (sqrt (+ t 1.0)) (* y 0.5)))
       (+ (sqrt t) (+ (sqrt z) (sqrt y)))))
     (+
      t_1
      (+
       (- (sqrt (+ y 1.0)) (sqrt y))
       (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + ((2.0 + (sqrt((t + 1.0)) + (y * 0.5))) - (sqrt(t) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t <= 170000000000.0d0) then
        tmp = t_1 + ((2.0d0 + (sqrt((t + 1.0d0)) + (y * 0.5d0))) - (sqrt(t) + (sqrt(z) + sqrt(y))))
    else
        tmp = t_1 + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + ((2.0 + (Math.sqrt((t + 1.0)) + (y * 0.5))) - (Math.sqrt(t) + (Math.sqrt(z) + Math.sqrt(y))));
	} else {
		tmp = t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t <= 170000000000.0:
		tmp = t_1 + ((2.0 + (math.sqrt((t + 1.0)) + (y * 0.5))) - (math.sqrt(t) + (math.sqrt(z) + math.sqrt(y))))
	else:
		tmp = t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t <= 170000000000.0)
		tmp = Float64(t_1 + Float64(Float64(2.0 + Float64(sqrt(Float64(t + 1.0)) + Float64(y * 0.5))) - Float64(sqrt(t) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t <= 170000000000.0)
		tmp = t_1 + ((2.0 + (sqrt((t + 1.0)) + (y * 0.5))) - (sqrt(t) + (sqrt(z) + sqrt(y))));
	else
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 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[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 170000000000.0], N[(t$95$1 + N[(N[(2.0 + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e11

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l+ 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+l- 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around 0 58.3%

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

      1. associate--l+ 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in z around 0 30.8%

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

   if 1.7e11 < t

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. +-commutative 82.8%

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

      5. associate-+l- 49.1%

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

      6. +-commutative 49.1%

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

      7. +-commutative 49.1%

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

   3. Simplified 49.1%

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

      1. associate--r- 82.8%

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

      2. flip-- 83.0%

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

      3. flip-- 83.3%

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

      4. frac-add 83.3%

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

   6. Applied egg-rr 83.4%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in t around inf 87.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(2 + \left(\sqrt{t + 1} + y \cdot 0.5\right)\right) - \left(\sqrt{t} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;t\_1 + \left(2 + \left(\left(\sqrt{t + 1} + y \cdot 0.5\right) - \left(\sqrt{z} + \left(\sqrt{t} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t 170000000000.0)
     (+
      t_1
      (+
       2.0
       (- (+ (sqrt (+ t 1.0)) (* y 0.5)) (+ (sqrt z) (+ (sqrt t) (sqrt y))))))
     (+
      t_1
      (+
       (- (sqrt (+ y 1.0)) (sqrt y))
       (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + (2.0 + ((sqrt((t + 1.0)) + (y * 0.5)) - (sqrt(z) + (sqrt(t) + sqrt(y)))));
	} else {
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t <= 170000000000.0d0) then
        tmp = t_1 + (2.0d0 + ((sqrt((t + 1.0d0)) + (y * 0.5d0)) - (sqrt(z) + (sqrt(t) + sqrt(y)))))
    else
        tmp = t_1 + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t <= 170000000000.0) {
		tmp = t_1 + (2.0 + ((Math.sqrt((t + 1.0)) + (y * 0.5)) - (Math.sqrt(z) + (Math.sqrt(t) + Math.sqrt(y)))));
	} else {
		tmp = t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t <= 170000000000.0:
		tmp = t_1 + (2.0 + ((math.sqrt((t + 1.0)) + (y * 0.5)) - (math.sqrt(z) + (math.sqrt(t) + math.sqrt(y)))))
	else:
		tmp = t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t <= 170000000000.0)
		tmp = Float64(t_1 + Float64(2.0 + Float64(Float64(sqrt(Float64(t + 1.0)) + Float64(y * 0.5)) - Float64(sqrt(z) + Float64(sqrt(t) + sqrt(y))))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t <= 170000000000.0)
		tmp = t_1 + (2.0 + ((sqrt((t + 1.0)) + (y * 0.5)) - (sqrt(z) + (sqrt(t) + sqrt(y)))));
	else
		tmp = t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 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[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 170000000000.0], N[(t$95$1 + N[(2.0 + N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e11

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l+ 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+l- 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around 0 58.3%

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

      1. associate--l+ 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in z around 0 30.8%

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

      1. associate--l+ 30.8%

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

      2. *-commutative 30.8%

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

      3. associate-+r+ 30.8%

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

      4. +-commutative 30.8%

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

   10. Simplified 30.8%

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

   if 1.7e11 < t

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. +-commutative 82.8%

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

      5. associate-+l- 49.1%

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

      6. +-commutative 49.1%

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

      7. +-commutative 49.1%

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

   3. Simplified 49.1%

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

      1. associate--r- 82.8%

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

      2. flip-- 83.0%

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

      3. flip-- 83.3%

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

      4. frac-add 83.3%

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

   6. Applied egg-rr 83.4%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in t around inf 87.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 170000000000:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(2 + \left(\left(\sqrt{t + 1} + y \cdot 0.5\right) - \left(\sqrt{z} + \left(\sqrt{t} + \sqrt{y}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \frac{1}{\sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.35e+14)
   (+
    (- (sqrt (+ x 1.0)) (sqrt x))
    (+ (- (sqrt (+ y 1.0)) (sqrt y)) (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0))))))
   (+
    (+ (* 0.5 (sqrt (/ 1.0 z))) (- (sqrt (+ t 1.0)) (sqrt t)))
    (+ (* 0.5 (sqrt (/ 1.0 y))) (* 0.5 (/ 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 (x <= 1.35e+14) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0)))));
	} else {
		tmp = ((0.5 * sqrt((1.0 / z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((0.5 * sqrt((1.0 / y))) + (0.5 * (1.0 / sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.35d+14) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0)))))
    else
        tmp = ((0.5d0 * sqrt((1.0d0 / z))) + (sqrt((t + 1.0d0)) - sqrt(t))) + ((0.5d0 * sqrt((1.0d0 / y))) + (0.5d0 * (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 (x <= 1.35e+14) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0)))));
	} else {
		tmp = ((0.5 * Math.sqrt((1.0 / z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) + ((0.5 * Math.sqrt((1.0 / y))) + (0.5 * (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 x <= 1.35e+14:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0)))))
	else:
		tmp = ((0.5 * math.sqrt((1.0 / z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) + ((0.5 * math.sqrt((1.0 / y))) + (0.5 * (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 (x <= 1.35e+14)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0))))));
	else
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(0.5 * Float64(1.0 / sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.35e+14)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 / (sqrt(z) + sqrt((z + 1.0)))));
	else
		tmp = ((0.5 * sqrt((1.0 / z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((0.5 * sqrt((1.0 / y))) + (0.5 * (1.0 / sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.35e+14], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e14

   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- 80.6%

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

      6. +-commutative 80.6%

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

      7. +-commutative 80.6%

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

   3. Simplified 80.6%

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

      1. associate--r- 97.3%

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

      2. flip-- 97.3%

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

      3. flip-- 97.5%

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

      4. frac-add 97.5%

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

   6. Applied egg-rr 98.0%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in t around inf 60.8%

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

   if 1.35e14 < x

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. associate-+l- 43.2%

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

      3. associate-+l- 83.9%

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

      4. +-commutative 83.9%

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

      5. +-commutative 83.9%

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

      6. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in x around inf 85.6%

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

   6. Taylor expanded in y around inf 40.9%

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

   7. Taylor expanded in z around inf 29.7%

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

      1. sqrt-div 29.7%

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

      2. metadata-eval 29.7%

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

   9. Applied egg-rr 29.7%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \frac{1}{\sqrt{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.95 \cdot 10^{-17}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + \left(\left(\left(y \cdot 0.5 - \sqrt{y}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 1.95e-17)
     (+
      (- t_1 (sqrt x))
      (+ (+ (- (* y 0.5) (sqrt y)) 1.0) (- (sqrt (+ z 1.0)) (sqrt z))))
     (if (<= y 1.25e+15)
       (+
        t_1
        (-
         (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z))))
         (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ 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((x + 1.0));
	double tmp;
	if (y <= 1.95e-17) {
		tmp = (t_1 - sqrt(x)) + ((((y * 0.5) - sqrt(y)) + 1.0) + (sqrt((z + 1.0)) - sqrt(z)));
	} else if (y <= 1.25e+15) {
		tmp = t_1 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (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((x + 1.0d0))
    if (y <= 1.95d-17) then
        tmp = (t_1 - sqrt(x)) + ((((y * 0.5d0) - sqrt(y)) + 1.0d0) + (sqrt((z + 1.0d0)) - sqrt(z)))
    else if (y <= 1.25d+15) then
        tmp = t_1 + ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (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((x + 1.0));
	double tmp;
	if (y <= 1.95e-17) {
		tmp = (t_1 - Math.sqrt(x)) + ((((y * 0.5) - Math.sqrt(y)) + 1.0) + (Math.sqrt((z + 1.0)) - Math.sqrt(z)));
	} else if (y <= 1.25e+15) {
		tmp = t_1 + ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (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((x + 1.0))
	tmp = 0
	if y <= 1.95e-17:
		tmp = (t_1 - math.sqrt(x)) + ((((y * 0.5) - math.sqrt(y)) + 1.0) + (math.sqrt((z + 1.0)) - math.sqrt(z)))
	elif y <= 1.25e+15:
		tmp = t_1 + ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (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(x + 1.0))
	tmp = 0.0
	if (y <= 1.95e-17)
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(Float64(Float64(y * 0.5) - sqrt(y)) + 1.0) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))));
	elseif (y <= 1.25e+15)
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 1.95e-17)
		tmp = (t_1 - sqrt(x)) + ((((y * 0.5) - sqrt(y)) + 1.0) + (sqrt((z + 1.0)) - sqrt(z)));
	elseif (y <= 1.25e+15)
		tmp = t_1 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.95e-17], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+15], N[(t$95$1 + 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[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.94999999999999995e-17

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. +-commutative 98.3%

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

      5. associate-+l- 79.3%

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

      6. +-commutative 79.3%

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

      7. +-commutative 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y around 0 79.3%

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

      1. associate--l+ 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in t around inf 53.7%

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

   if 1.94999999999999995e-17 < y < 1.25e15

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. associate-+l- 67.4%

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

      3. associate-+l- 83.6%

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

      4. +-commutative 83.6%

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

      5. +-commutative 83.6%

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

      6. +-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in t around inf 23.1%

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

      1. associate--l+ 32.2%

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

      2. +-commutative 32.2%

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

      3. +-commutative 32.2%

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

   7. Simplified 32.2%

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

   8. Taylor expanded in z around inf 34.2%

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

   if 1.25e15 < y

   1. Initial program 82.5%

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

      1. associate-+l+ 82.5%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in t around inf 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+ 24.5%

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

      2. +-commutative 24.5%

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

      3. +-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in x around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cube-cbrt 25.1%

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

       2. pow3 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 24.1%

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

       2. flip-+ 24.0%

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

       3. add-sqr-sqrt 24.4%

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

       4. pow2 24.4%

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

   14. Applied egg-rr 24.4%

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

       1. associate--l+ 26.2%

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

       2. unpow2 26.2%

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

       3. sqr-neg 26.2%

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

       4. rem-square-sqrt 27.1%

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

       5. +-inverses 27.1%

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

       6. metadata-eval 27.1%

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

       7. sub-neg 27.1%

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

       8. remove-double-neg 27.1%

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

       9. +-commutative 27.1%

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

   16. Simplified 27.1%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(y \cdot 0.5 - \sqrt{y}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.36 \cdot 10^{-17}:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right) + 1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;t\_2 + \left(\left(t\_1 + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= y 1.36e-17)
     (+ (+ t_1 (- (sqrt (+ z 1.0)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))) 1.0)
     (if (<= y 3.7e+15)
       (+ t_2 (- (+ t_1 (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ t_2 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (y <= 1.36e-17) {
		tmp = (t_1 + (sqrt((z + 1.0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0;
	} else if (y <= 3.7e+15) {
		tmp = t_2 + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (t_2 + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (y <= 1.36d-17) then
        tmp = (t_1 + (sqrt((z + 1.0d0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0d0
    else if (y <= 3.7d+15) then
        tmp = t_2 + ((t_1 + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (t_2 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 1.36e-17) {
		tmp = (t_1 + (Math.sqrt((z + 1.0)) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))))) + 1.0;
	} else if (y <= 3.7e+15) {
		tmp = t_2 + ((t_1 + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (t_2 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 1.36e-17:
		tmp = (t_1 + (math.sqrt((z + 1.0)) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))) + 1.0
	elif y <= 3.7e+15:
		tmp = t_2 + ((t_1 + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (t_2 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.36e-17)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))))) + 1.0);
	elseif (y <= 3.7e+15)
		tmp = Float64(t_2 + Float64(Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(t_2 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 1.36e-17)
		tmp = (t_1 + (sqrt((z + 1.0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0;
	elseif (y <= 3.7e+15)
		tmp = t_2 + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (t_2 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.36e-17], N[(N[(t$95$1 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 3.7e+15], N[(t$95$2 + N[(N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.36e-17

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 61.5%

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

      3. associate-+l- 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t around inf 21.1%

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

      1. associate--l+ 24.2%

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

      2. +-commutative 24.2%

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

      3. +-commutative 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in x around 0 17.0%

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

      1. associate--l+ 24.8%

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

      2. associate--l+ 32.4%

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

   10. Simplified 32.4%

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

   if 1.36e-17 < y < 3.7e15

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. associate-+l- 64.3%

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

      3. associate-+l- 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. +-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in t around inf 21.7%

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

      1. associate--l+ 30.3%

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

      2. +-commutative 30.3%

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

      3. +-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in z around inf 32.6%

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

   if 3.7e15 < y

   1. Initial program 82.5%

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

      1. associate-+l+ 82.5%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in t around inf 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+ 24.5%

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

      2. +-commutative 24.5%

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

      3. +-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in x around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cube-cbrt 25.1%

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

       2. pow3 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 24.1%

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

       2. flip-+ 24.0%

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

       3. add-sqr-sqrt 24.4%

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

       4. pow2 24.4%

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

   14. Applied egg-rr 24.4%

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

       1. associate--l+ 26.2%

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

       2. unpow2 26.2%

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

       3. sqr-neg 26.2%

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

       4. rem-square-sqrt 27.1%

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

       5. +-inverses 27.1%

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

       6. metadata-eval 27.1%

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

       7. sub-neg 27.1%

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

       8. remove-double-neg 27.1%

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

       9. +-commutative 27.1%

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

   16. Simplified 27.1%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right) + 1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right) + 1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t\_2 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= y 7.2e-19)
     (+ (+ t_1 (- (sqrt (+ z 1.0)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))) 1.0)
     (if (<= y 5e+15)
       (+ t_2 (- t_1 (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ t_2 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (y <= 7.2e-19) {
		tmp = (t_1 + (sqrt((z + 1.0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0;
	} else if (y <= 5e+15) {
		tmp = t_2 + (t_1 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (t_2 + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (y <= 7.2d-19) then
        tmp = (t_1 + (sqrt((z + 1.0d0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0d0
    else if (y <= 5d+15) then
        tmp = t_2 + (t_1 - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (t_2 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 7.2e-19) {
		tmp = (t_1 + (Math.sqrt((z + 1.0)) - (Math.sqrt(x) + (Math.sqrt(z) + Math.sqrt(y))))) + 1.0;
	} else if (y <= 5e+15) {
		tmp = t_2 + (t_1 - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (t_2 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 7.2e-19:
		tmp = (t_1 + (math.sqrt((z + 1.0)) - (math.sqrt(x) + (math.sqrt(z) + math.sqrt(y))))) + 1.0
	elif y <= 5e+15:
		tmp = t_2 + (t_1 - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (t_2 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 7.2e-19)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))))) + 1.0);
	elseif (y <= 5e+15)
		tmp = Float64(t_2 + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(t_2 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 7.2e-19)
		tmp = (t_1 + (sqrt((z + 1.0)) - (sqrt(x) + (sqrt(z) + sqrt(y))))) + 1.0;
	elseif (y <= 5e+15)
		tmp = t_2 + (t_1 - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (t_2 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 7.2e-19], N[(N[(t$95$1 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 5e+15], N[(t$95$2 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000002e-19

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 61.8%

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

      3. associate-+l- 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

      6. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t around inf 21.2%

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

      1. associate--l+ 24.4%

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

      2. +-commutative 24.4%

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

      3. +-commutative 24.4%

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

   7. Simplified 24.4%

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

   8. Taylor expanded in x around 0 17.1%

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

      1. associate--l+ 25.0%

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

      2. associate--l+ 32.7%

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

   10. Simplified 32.7%

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

   if 7.2000000000000002e-19 < y < 5e15

   1. Initial program 85.7%

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

      1. associate-+l+ 85.7%

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

      2. associate-+l- 61.5%

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

      3. associate-+l- 85.7%

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

      4. +-commutative 85.7%

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

      5. +-commutative 85.7%

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

      6. +-commutative 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in t around inf 20.6%

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

      1. associate--l+ 28.6%

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

      2. +-commutative 28.6%

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

      3. +-commutative 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in z around inf 32.7%

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

   if 5e15 < y

   1. Initial program 82.5%

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

      1. associate-+l+ 82.5%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in t around inf 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+ 24.5%

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

      2. +-commutative 24.5%

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

      3. +-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in x around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cube-cbrt 25.1%

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

       2. pow3 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 24.1%

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

       2. flip-+ 24.0%

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

       3. add-sqr-sqrt 24.4%

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

       4. pow2 24.4%

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

   14. Applied egg-rr 24.4%

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

       1. associate--l+ 26.2%

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

       2. unpow2 26.2%

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

       3. sqr-neg 26.2%

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

       4. rem-square-sqrt 27.1%

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

       5. +-inverses 27.1%

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

       6. metadata-eval 27.1%

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

       7. sub-neg 27.1%

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

       8. remove-double-neg 27.1%

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

       9. +-commutative 27.1%

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

   16. Simplified 27.1%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\right) + 1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 1.45:\\ \;\;\;\;t\_2 + \left(\left(t\_3 + 1\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_3 - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ y 1.0))))
   (if (<= z 1.45) (+ t_2 (- (+ t_3 1.0) t_1)) (+ t_2 (- t_3 t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((y + 1.0));
	double tmp;
	if (z <= 1.45) {
		tmp = t_2 + ((t_3 + 1.0) - t_1);
	} else {
		tmp = t_2 + (t_3 - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((y + 1.0d0))
    if (z <= 1.45d0) then
        tmp = t_2 + ((t_3 + 1.0d0) - t_1)
    else
        tmp = t_2 + (t_3 - 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(x) + Math.sqrt(y);
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 1.45) {
		tmp = t_2 + ((t_3 + 1.0) - t_1);
	} else {
		tmp = t_2 + (t_3 - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 1.45:
		tmp = t_2 + ((t_3 + 1.0) - t_1)
	else:
		tmp = t_2 + (t_3 - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 1.45)
		tmp = Float64(t_2 + Float64(Float64(t_3 + 1.0) - t_1));
	else
		tmp = Float64(t_2 + Float64(t_3 - t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 1.45)
		tmp = t_2 + ((t_3 + 1.0) - t_1);
	else
		tmp = t_2 + (t_3 - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.45], N[(t$95$2 + N[(N[(t$95$3 + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$3 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.44999999999999996

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. associate-+l- 75.2%

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

      3. associate-+l- 97.9%

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

      4. +-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t around inf 23.5%

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

      1. associate--l+ 26.2%

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

      2. +-commutative 26.2%

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

      3. +-commutative 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in y around inf 26.0%

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

   9. Taylor expanded in z around 0 26.0%

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

   if 1.44999999999999996 < z

   1. Initial program 84.6%

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

      1. associate-+l+ 84.6%

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

      2. associate-+l- 66.9%

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

      3. associate-+l- 84.6%

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

      4. +-commutative 84.6%

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

      5. +-commutative 84.6%

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

      6. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in t around inf 5.4%

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

      1. associate--l+ 23.3%

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

      2. +-commutative 23.3%

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

      3. +-commutative 23.3%

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

   7. Simplified 23.3%

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

   8. Taylor expanded in z around inf 33.1%

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

4. Final simplification 29.8%

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

Alternative 16: 70.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1 + \left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 4.5e+15)
     (+ t_1 (- (sqrt (+ y 1.0)) (+ (sqrt x) (sqrt y))))
     (/ 1.0 (+ 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((x + 1.0));
	double tmp;
	if (y <= 4.5e+15) {
		tmp = t_1 + (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (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((x + 1.0d0))
    if (y <= 4.5d+15) then
        tmp = t_1 + (sqrt((y + 1.0d0)) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (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((x + 1.0));
	double tmp;
	if (y <= 4.5e+15) {
		tmp = t_1 + (Math.sqrt((y + 1.0)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (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((x + 1.0))
	tmp = 0
	if y <= 4.5e+15:
		tmp = t_1 + (math.sqrt((y + 1.0)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (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(x + 1.0))
	tmp = 0.0
	if (y <= 4.5e+15)
		tmp = Float64(t_1 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 4.5e+15)
		tmp = t_1 + (sqrt((y + 1.0)) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.5e+15], N[(t$95$1 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e15

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l- 61.8%

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

      3. associate-+l- 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

      6. +-commutative 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t around inf 21.2%

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

      1. associate--l+ 24.8%

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

      2. +-commutative 24.8%

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

      3. +-commutative 24.8%

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

   7. Simplified 24.8%

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

   8. Taylor expanded in z around inf 23.1%

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

   if 4.5e15 < y

   1. Initial program 82.5%

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

      1. associate-+l+ 82.5%

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

      2. associate-+l- 82.5%

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

      3. associate-+l- 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in t around inf 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+ 24.5%

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

      2. +-commutative 24.5%

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

      3. +-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in x around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cube-cbrt 25.1%

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

       2. pow3 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 24.1%

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

       2. flip-+ 24.0%

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

       3. add-sqr-sqrt 24.4%

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

       4. pow2 24.4%

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

   14. Applied egg-rr 24.4%

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

       1. associate--l+ 26.2%

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

       2. unpow2 26.2%

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

       3. sqr-neg 26.2%

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

       4. rem-square-sqrt 27.1%

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

       5. +-inverses 27.1%

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

       6. metadata-eval 27.1%

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

       7. sub-neg 27.1%

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

       8. remove-double-neg 27.1%

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

       9. +-commutative 27.1%

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

   16. Simplified 27.1%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 (<= x 58000000.0) (- (sqrt (+ x 1.0)) (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 58000000.0d0) then
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 58000000.0:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 58000000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 58000000.0)
		tmp = sqrt((x + 1.0)) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 58000000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e7

   1. Initial program 97.6%

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

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

      3. associate-+l- 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t around inf 21.0%

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

      1. associate--l+ 42.1%

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

      2. +-commutative 42.1%

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

      3. +-commutative 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in x around inf 30.5%

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

      1. mul-1-neg 30.5%

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

   10. Simplified 30.5%

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

       1. unsub-neg 30.5%

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

   12. Applied egg-rr 30.5%

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

   if 5.8e7 < x

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. associate-+l- 43.8%

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

      3. associate-+l- 83.9%

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

      4. +-commutative 83.9%

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

      5. +-commutative 83.9%

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

      6. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in t around inf 6.5%

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

      1. associate--l+ 7.3%

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

      2. +-commutative 7.3%

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

      3. +-commutative 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in x around inf 3.3%

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

      1. mul-1-neg 3.3%

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

   10. Simplified 3.3%

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

   11. Taylor expanded in x around inf 7.8%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.0% accurate, 4.0× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ 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) {
	return 1.0 / (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 = 1.0d0 / (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 1.0 / (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 1.0 / (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(1.0 / 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 = 1.0 / (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[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

   1. associate-+l+ 90.7%

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

   2. associate-+l- 70.7%

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

   3. associate-+l- 90.7%

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

   4. +-commutative 90.7%

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

   5. +-commutative 90.7%

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

   6. +-commutative 90.7%

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

3. Simplified 90.7%

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

5. Taylor expanded in t around inf 13.7%

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

   1. associate--l+ 24.7%

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

   2. +-commutative 24.7%

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

   3. +-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in x around inf 16.9%

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

   1. mul-1-neg 16.9%

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

10. Simplified 16.9%

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

    1. add-cube-cbrt 17.9%

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

    2. pow3 17.9%

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

12. Applied egg-rr 17.9%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}}\right)}^{3}} + \left(-\sqrt{x}\right) \]
13. Step-by-step derivation

    1. rem-cube-cbrt 16.9%

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

    2. flip-+ 16.9%

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

    3. add-sqr-sqrt 17.1%

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

    4. pow2 17.1%

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

14. Applied egg-rr 17.1%

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

    1. associate--l+ 18.4%

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

    2. unpow2 18.4%

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

    3. sqr-neg 18.4%

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

    4. rem-square-sqrt 19.1%

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

    5. +-inverses 19.1%

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

    6. metadata-eval 19.1%

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

    7. sub-neg 19.1%

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

    8. remove-double-neg 19.1%

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

    9. +-commutative 19.1%

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

16. Simplified 19.1%

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

17. Final simplification 19.1%

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
18. Add Preprocessing

Alternative 19: 39.5% accurate, 7.3× speedup

Math

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

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 (<= x 1.0) (+ (- (* 0.5 x) (sqrt x)) 1.0) (* 0.5 (sqrt (/ 1.0 x)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l- 97.7%

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

      3. associate-+l- 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t around inf 21.1%

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

      1. associate--l+ 42.6%

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

      2. +-commutative 42.6%

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

      3. +-commutative 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in x around inf 30.3%

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

      1. mul-1-neg 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in x around 0 30.3%

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

       1. associate--l+ 30.3%

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

   13. Simplified 30.3%

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

   if 1 < x

   1. Initial program 84.1%

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

      1. associate-+l+ 84.1%

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

      2. associate-+l- 44.9%

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

      3. associate-+l- 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

      6. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in t around inf 6.7%

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

      1. associate--l+ 7.5%

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

      2. +-commutative 7.5%

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

      3. +-commutative 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in x around inf 4.1%

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

      1. mul-1-neg 4.1%

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

   10. Simplified 4.1%

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

   11. Taylor expanded in x around inf 8.0%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.9%

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

Alternative 20: 39.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.021:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 (<= x 0.021) (- 1.0 (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.021) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 0.021d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.021) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 0.021:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 0.021)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 0.021)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 0.021], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0210000000000000013

   1. Initial program 97.7%

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

      1. associate-+l+ 97.7%

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

      2. associate-+l- 97.7%

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

      3. associate-+l- 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t around inf 21.1%

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

      1. associate--l+ 42.6%

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

      2. +-commutative 42.6%

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

      3. +-commutative 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in x around inf 30.3%

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

      1. mul-1-neg 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in x around 0 30.0%

       \[\leadsto \color{blue}{1 - \sqrt{x}} \]

   if 0.0210000000000000013 < x

   1. Initial program 84.1%

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

      1. associate-+l+ 84.1%

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

      2. associate-+l- 44.9%

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

      3. associate-+l- 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

      6. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in t around inf 6.7%

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

      1. associate--l+ 7.5%

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

      2. +-commutative 7.5%

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

      3. +-commutative 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in x around inf 4.1%

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

      1. mul-1-neg 4.1%

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

   10. Simplified 4.1%

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

   11. Taylor expanded in x around inf 8.0%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.1% 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 90.7%

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

   1. associate-+l+ 90.7%

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

   2. associate-+l- 70.7%

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

   3. associate-+l- 90.7%

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

   4. +-commutative 90.7%

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

   5. +-commutative 90.7%

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

   6. +-commutative 90.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 90.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 13.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 24.7%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. +-commutative 24.7%

      \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in x around inf 16.9%

   \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 16.9%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

10. Simplified 16.9%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

11. Taylor expanded in x around 0 15.3%

    \[\leadsto \color{blue}{1 - \sqrt{x}} \]
12. Add Preprocessing

Alternative 22: 1.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := (-N[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 90.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. associate-+l- 70.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+l- 90.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 90.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 90.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 90.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 90.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 13.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 24.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 24.7%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. +-commutative 24.7%

      \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in x around inf 16.9%

   \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 16.9%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

10. Simplified 16.9%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

11. Taylor expanded in x around 0 15.3%

    \[\leadsto \color{blue}{1 - \sqrt{x}} \]

12. Taylor expanded in x around inf 1.6%

    \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} \]
13. Step-by-step derivation

    1. neg-mul-1 1.6%

       \[\leadsto \color{blue}{-\sqrt{x}} \]

14. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]
15. Add Preprocessing

Developer Target 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024116 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))