Main:z from

Percentage Accurate: 91.7% → 98.8%

Time: 42.6s

Alternatives: 22

Speedup: 1.6×

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: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\left(t\_2 - \sqrt{z}\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + {\left(\sqrt{t\_1}\right)}^{2}\right) + \left(t\_3 + \frac{1}{t\_2 + \sqrt{z}}\right)\\ \end{array} \end{array} \]

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)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 5e-7)
     (+
      (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (* 0.5 (sqrt (/ 1.0 y))))
      (+ (- t_2 (sqrt z)) t_3))
     (+
      (+ (- 1.0 (sqrt x)) (pow (sqrt t_1) 2.0))
      (+ t_3 (/ 1.0 (+ t_2 (sqrt z))))))))

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)) - sqrt(y);
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (0.5 * sqrt((1.0 / y)))) + ((t_2 - sqrt(z)) + t_3);
	} else {
		tmp = ((1.0 - sqrt(x)) + pow(sqrt(t_1), 2.0)) + (t_3 + (1.0 / (t_2 + 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0)) - sqrt(y)
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 5d-7) then
        tmp = ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (0.5d0 * sqrt((1.0d0 / y)))) + ((t_2 - sqrt(z)) + t_3)
    else
        tmp = ((1.0d0 - sqrt(x)) + (sqrt(t_1) ** 2.0d0)) + (t_3 + (1.0d0 / (t_2 + 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((y + 1.0)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / y)))) + ((t_2 - Math.sqrt(z)) + t_3);
	} else {
		tmp = ((1.0 - Math.sqrt(x)) + Math.pow(Math.sqrt(t_1), 2.0)) + (t_3 + (1.0 / (t_2 + 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((y + 1.0)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 5e-7:
		tmp = ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (0.5 * math.sqrt((1.0 / y)))) + ((t_2 - math.sqrt(z)) + t_3)
	else:
		tmp = ((1.0 - math.sqrt(x)) + math.pow(math.sqrt(t_1), 2.0)) + (t_3 + (1.0 / (t_2 + 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(y + 1.0)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(Float64(t_2 - sqrt(z)) + t_3));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + (sqrt(t_1) ^ 2.0)) + Float64(t_3 + Float64(1.0 / Float64(t_2 + 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((y + 1.0)) - sqrt(y);
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 5e-7)
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (0.5 * sqrt((1.0 / y)))) + ((t_2 - sqrt(z)) + t_3);
	else
		tmp = ((1.0 - sqrt(x)) + (sqrt(t_1) ^ 2.0)) + (t_3 + (1.0 / (t_2 + 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[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-7], N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $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)) < 4.99999999999999977e-7

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

   3. Simplified 86.7%

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

      1. flip-- 86.7%

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

      2. add-sqr-sqrt 74.5%

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

      3. +-commutative 74.5%

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

      4. add-sqr-sqrt 87.0%

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

      5. +-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate--l+ 89.9%

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

      2. +-inverses 89.9%

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

      3. metadata-eval 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in y around inf 92.1%

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

   if 4.99999999999999977e-7 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. +-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 x around 0 48.9%

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

      1. flip-- 48.9%

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

      2. add-sqr-sqrt 39.3%

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

      3. add-sqr-sqrt 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate--l+ 49.2%

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

      2. +-inverses 49.2%

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

      3. metadata-eval 49.2%

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

      4. +-commutative 49.2%

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

   9. Simplified 49.2%

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

       1. add-sqr-sqrt 49.2%

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

       2. pow2 49.2%

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

   11. Applied egg-rr 49.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + {\left(\sqrt{\sqrt{y + 1} - \sqrt{y}}\right)}^{2}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\left(t\_2 - \sqrt{z}\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \frac{1}{t\_2 + \sqrt{z}}\right) + \left(\left(1 - \sqrt{x}\right) + t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 5e-7)
     (+
      (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (* 0.5 (sqrt (/ 1.0 y))))
      (+ (- t_2 (sqrt z)) t_3))
     (+ (+ t_3 (/ 1.0 (+ t_2 (sqrt z)))) (+ (- 1.0 (sqrt x)) t_1)))))

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)) - sqrt(y);
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (0.5 * sqrt((1.0 / y)))) + ((t_2 - sqrt(z)) + t_3);
	} else {
		tmp = (t_3 + (1.0 / (t_2 + sqrt(z)))) + ((1.0 - sqrt(x)) + 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((y + 1.0d0)) - sqrt(y)
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 5d-7) then
        tmp = ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (0.5d0 * sqrt((1.0d0 / y)))) + ((t_2 - sqrt(z)) + t_3)
    else
        tmp = (t_3 + (1.0d0 / (t_2 + sqrt(z)))) + ((1.0d0 - sqrt(x)) + 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((y + 1.0)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / y)))) + ((t_2 - Math.sqrt(z)) + t_3);
	} else {
		tmp = (t_3 + (1.0 / (t_2 + Math.sqrt(z)))) + ((1.0 - Math.sqrt(x)) + t_1);
	}
	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)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 5e-7:
		tmp = ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (0.5 * math.sqrt((1.0 / y)))) + ((t_2 - math.sqrt(z)) + t_3)
	else:
		tmp = (t_3 + (1.0 / (t_2 + math.sqrt(z)))) + ((1.0 - math.sqrt(x)) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(Float64(t_2 - sqrt(z)) + t_3));
	else
		tmp = Float64(Float64(t_3 + Float64(1.0 / Float64(t_2 + sqrt(z)))) + Float64(Float64(1.0 - sqrt(x)) + 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((y + 1.0)) - sqrt(y);
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 5e-7)
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (0.5 * sqrt((1.0 / y)))) + ((t_2 - sqrt(z)) + t_3);
	else
		tmp = (t_3 + (1.0 / (t_2 + sqrt(z)))) + ((1.0 - sqrt(x)) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 4.99999999999999977e-7

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

   3. Simplified 86.7%

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

      1. flip-- 86.7%

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

      2. add-sqr-sqrt 74.5%

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

      3. +-commutative 74.5%

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

      4. add-sqr-sqrt 87.0%

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

      5. +-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate--l+ 89.9%

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

      2. +-inverses 89.9%

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

      3. metadata-eval 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in y around inf 92.1%

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

   if 4.99999999999999977e-7 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. +-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 x around 0 48.9%

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

      1. flip-- 48.9%

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

      2. add-sqr-sqrt 39.3%

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

      3. add-sqr-sqrt 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate--l+ 49.2%

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

      2. +-inverses 49.2%

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

      3. metadata-eval 49.2%

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

      4. +-commutative 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;y \leq 56000000000000:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t\_1 + \sqrt{z}}\right) + \left(\left(1 - \sqrt{x}\right) + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + t\_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= y 56000000000000.0)
     (+
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ t_1 (sqrt z))))
      (+ (- 1.0 (sqrt x)) t_2))
     (+ (- t_1 (sqrt z)) (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (y <= 56000000000000.0) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z)))) + ((1.0 - sqrt(x)) + t_2);
	} else {
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    if (y <= 56000000000000.0d0) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (t_1 + sqrt(z)))) + ((1.0d0 - sqrt(x)) + t_2)
    else
        tmp = (t_1 - sqrt(z)) + ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (y <= 56000000000000.0) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (t_1 + Math.sqrt(z)))) + ((1.0 - Math.sqrt(x)) + t_2);
	} else {
		tmp = (t_1 - Math.sqrt(z)) + ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if y <= 56000000000000.0:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (t_1 + math.sqrt(z)))) + ((1.0 - math.sqrt(x)) + t_2)
	else:
		tmp = (t_1 - math.sqrt(z)) + ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (y <= 56000000000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(t_1 + sqrt(z)))) + Float64(Float64(1.0 - sqrt(x)) + t_2));
	else
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (y <= 56000000000000.0)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (t_1 + sqrt(z)))) + ((1.0 - sqrt(x)) + t_2);
	else
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 56000000000000.0], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6e13

   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. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. +-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 x around 0 48.9%

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

      1. flip-- 48.9%

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

      2. add-sqr-sqrt 39.3%

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

      3. add-sqr-sqrt 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate--l+ 49.2%

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

      2. +-inverses 49.2%

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

      3. metadata-eval 49.2%

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

      4. +-commutative 49.2%

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

   9. Simplified 49.2%

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

   if 5.6e13 < y

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. +-commutative 86.7%

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

   3. Simplified 86.7%

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

      1. flip-- 86.7%

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

      2. add-sqr-sqrt 74.5%

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

      3. +-commutative 74.5%

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

      4. add-sqr-sqrt 87.0%

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

      5. +-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate--l+ 89.9%

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

      2. +-inverses 89.9%

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

      3. metadata-eval 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in t around inf 50.4%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 56000000000000:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(t\_1 + 2\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= t 1.9e+25)
     (-
      (+ (+ t_1 2.0) (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))))
      (+ (sqrt y) (+ (sqrt x) (sqrt z))))
     (+
      (- t_1 (sqrt z))
      (+
       (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
       (- (sqrt (+ y 1.0)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (t <= 1.9e+25) {
		tmp = ((t_1 + 2.0) + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))) - (sqrt(y) + (sqrt(x) + sqrt(z)));
	} else {
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (sqrt((y + 1.0)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 1.9d+25) then
        tmp = ((t_1 + 2.0d0) + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t)))) - (sqrt(y) + (sqrt(x) + sqrt(z)))
    else
        tmp = (t_1 - sqrt(z)) + ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (sqrt((y + 1.0d0)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (t <= 1.9e+25) {
		tmp = ((t_1 + 2.0) + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z)));
	} else {
		tmp = (t_1 - Math.sqrt(z)) + ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if t <= 1.9e+25:
		tmp = ((t_1 + 2.0) + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z)))
	else:
		tmp = (t_1 - math.sqrt(z)) + ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (math.sqrt((y + 1.0)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t <= 1.9e+25)
		tmp = Float64(Float64(Float64(t_1 + 2.0) + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z))));
	else
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t <= 1.9e+25)
		tmp = ((t_1 + 2.0) + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))) - (sqrt(y) + (sqrt(x) + sqrt(z)));
	else
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (sqrt((y + 1.0)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.9e+25], N[(N[(N[(t$95$1 + 2.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9e25

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 76.7%

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

      4. associate-+l- 57.9%

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

      5. associate-+r- 54.4%

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

   3. Simplified 44.7%

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

      1. flip-- 44.7%

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

      2. add-sqr-sqrt 44.5%

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

      3. add-sqr-sqrt 44.7%

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

      4. +-commutative 44.7%

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

      5. +-commutative 44.7%

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

   6. Applied egg-rr 44.7%

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

      1. associate--r+ 45.2%

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

      2. div-sub 45.2%

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

      3. +-inverses 45.2%

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

      4. metadata-eval 45.2%

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

      5. +-inverses 45.2%

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

      6. sub-neg 45.2%

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

      7. associate-+l+ 44.7%

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

      8. +-commutative 44.7%

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

      9. sub-neg 44.7%

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

      10. div-sub 44.7%

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

      11. neg-sub0 44.7%

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

      12. sub-neg 44.7%

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

      13. +-commutative 44.7%

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

      14. associate-+l+ 45.2%

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

      15. sub-neg 45.2%

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

      16. +-inverses 45.2%

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

      17. metadata-eval 45.2%

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

      18. metadata-eval 45.2%

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

   8. Simplified 45.2%

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

   9. Taylor expanded in x around 0 23.1%

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

   10. Taylor expanded in y around 0 16.9%

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

       1. associate-+r+ 16.9%

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

       2. +-commutative 16.9%

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

       3. +-commutative 16.9%

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

       4. associate-+r+ 16.9%

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

       5. +-commutative 16.9%

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

   12. Simplified 16.9%

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

   if 1.9e25 < t

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. +-commutative 86.9%

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

      3. +-commutative 86.9%

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

      4. +-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. flip-- 86.9%

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

      2. add-sqr-sqrt 72.2%

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

      3. +-commutative 72.2%

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

      4. add-sqr-sqrt 87.4%

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

      5. +-commutative 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. associate--l+ 90.2%

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

      2. +-inverses 90.2%

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

      3. metadata-eval 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in t around inf 90.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} + 2\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;t \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= t 1.9e+25)
     (+
      (+ t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))))
     (+
      t_1
      (+
       (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
       (- (sqrt (+ y 1.0)) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if (t <= 1.9e+25) {
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (sqrt((y + 1.0)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    if (t <= 1.9d+25) then
        tmp = (t_1 + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y)))
    else
        tmp = t_1 + ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (sqrt((y + 1.0d0)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if (t <= 1.9e+25) {
		tmp = (t_1 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	tmp = 0
	if t <= 1.9e+25:
		tmp = (t_1 + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y)))
	else:
		tmp = t_1 + ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (math.sqrt((y + 1.0)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (t <= 1.9e+25)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if (t <= 1.9e+25)
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	else
		tmp = t_1 + ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (sqrt((y + 1.0)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.9e+25], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9e25

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 53.3%

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

   6. Taylor expanded in y around 0 26.5%

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

   if 1.9e25 < t

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. +-commutative 86.9%

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

      3. +-commutative 86.9%

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

      4. +-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. flip-- 86.9%

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

      2. add-sqr-sqrt 72.2%

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

      3. +-commutative 72.2%

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

      4. add-sqr-sqrt 87.4%

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

      5. +-commutative 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. associate--l+ 90.2%

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

      2. +-inverses 90.2%

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

      3. metadata-eval 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in t around inf 90.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 - \sqrt{x}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(t\_1 + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+22}:\\ \;\;\;\;t\_2 + \left(t\_1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 (- 1.0 (sqrt x))) (t_2 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= y 2.8e-22)
     (+ (+ t_2 (- (sqrt (+ 1.0 t)) (sqrt t))) (+ t_1 (- 1.0 (sqrt y))))
     (if (<= y 9e+22)
       (+ t_2 (+ t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ y 1.0))))))
       (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - sqrt(x);
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if (y <= 2.8e-22) {
		tmp = (t_2 + (sqrt((1.0 + t)) - sqrt(t))) + (t_1 + (1.0 - sqrt(y)));
	} else if (y <= 9e+22) {
		tmp = t_2 + (t_1 + (1.0 / (sqrt(y) + sqrt((y + 1.0)))));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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 = 1.0d0 - sqrt(x)
    t_2 = sqrt((1.0d0 + z)) - sqrt(z)
    if (y <= 2.8d-22) then
        tmp = (t_2 + (sqrt((1.0d0 + t)) - sqrt(t))) + (t_1 + (1.0d0 - sqrt(y)))
    else if (y <= 9d+22) then
        tmp = t_2 + (t_1 + (1.0d0 / (sqrt(y) + sqrt((y + 1.0d0)))))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + 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 = 1.0 - Math.sqrt(x);
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double tmp;
	if (y <= 2.8e-22) {
		tmp = (t_2 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (t_1 + (1.0 - Math.sqrt(y)));
	} else if (y <= 9e+22) {
		tmp = t_2 + (t_1 + (1.0 / (Math.sqrt(y) + Math.sqrt((y + 1.0)))));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 - math.sqrt(x)
	t_2 = math.sqrt((1.0 + z)) - math.sqrt(z)
	tmp = 0
	if y <= 2.8e-22:
		tmp = (t_2 + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (t_1 + (1.0 - math.sqrt(y)))
	elif y <= 9e+22:
		tmp = t_2 + (t_1 + (1.0 / (math.sqrt(y) + math.sqrt((y + 1.0)))))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 - sqrt(x))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (y <= 2.8e-22)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(t_1 + Float64(1.0 - sqrt(y))));
	elseif (y <= 9e+22)
		tmp = Float64(t_2 + Float64(t_1 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + 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 = 1.0 - sqrt(x);
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	tmp = 0.0;
	if (y <= 2.8e-22)
		tmp = (t_2 + (sqrt((1.0 + t)) - sqrt(t))) + (t_1 + (1.0 - sqrt(y)));
	elseif (y <= 9e+22)
		tmp = t_2 + (t_1 + (1.0 / (sqrt(y) + sqrt((y + 1.0)))));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.8e-22], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+22], N[(t$95$2 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.79999999999999995e-22

   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. +-commutative 97.7%

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

      3. +-commutative 97.7%

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

      4. +-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 x around 0 49.0%

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

   6. Taylor expanded in y around 0 49.0%

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

   if 2.79999999999999995e-22 < y < 8.9999999999999996e22

   1. Initial program 94.6%

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

      1. associate-+l+ 94.6%

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

      2. +-commutative 94.6%

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

      3. +-commutative 94.6%

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

      4. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in x around 0 50.1%

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

   6. Taylor expanded in t around inf 27.0%

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

      1. flip-- 27.0%

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

      2. add-sqr-sqrt 26.9%

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

      3. add-sqr-sqrt 26.9%

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

   8. Applied egg-rr 26.9%

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

      1. associate--l+ 26.9%

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

      2. +-inverses 26.9%

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

      3. metadata-eval 26.9%

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

      4. +-commutative 26.9%

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

   10. Simplified 26.9%

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

   if 8.9999999999999996e22 < y

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-+r- 44.4%

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

      4. associate-+l- 23.6%

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

      5. associate-+r- 4.3%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 22.8%

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

      2. +-commutative 22.8%

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

      3. +-commutative 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around inf 22.2%

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

      1. mul-1-neg 22.2%

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

   10. Simplified 22.2%

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

       1. flip-+ 22.2%

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

       2. add-sqr-sqrt 22.3%

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

       3. pow2 22.3%

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

   12. Applied egg-rr 22.3%

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

       1. associate--l+ 24.0%

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

       2. unpow2 24.0%

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

       3. sqr-neg 24.0%

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

       4. rem-square-sqrt 26.0%

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

       5. sub-neg 26.0%

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

       6. remove-double-neg 26.0%

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

   14. Simplified 26.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.2e+22)
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
   (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.2d+22) then
        tmp = (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + ((1.0d0 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.2e+22:
		tmp = (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + ((1.0 - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.2e+22)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.2e+22)
		tmp = (1.0 / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.2e+22], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e22

   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. +-commutative 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 49.1%

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

      1. flip-- 49.1%

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

      2. add-sqr-sqrt 39.5%

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

      3. add-sqr-sqrt 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. associate--l+ 49.4%

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

      2. +-inverses 49.4%

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

      3. metadata-eval 49.4%

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

      4. +-commutative 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in t around inf 27.7%

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

   if 1.2e22 < y

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-+r- 44.4%

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

      4. associate-+l- 23.6%

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

      5. associate-+r- 4.3%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 22.8%

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

      2. +-commutative 22.8%

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

      3. +-commutative 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around inf 22.2%

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

      1. mul-1-neg 22.2%

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

   10. Simplified 22.2%

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

       1. flip-+ 22.2%

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

       2. add-sqr-sqrt 22.3%

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

       3. pow2 22.3%

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

   12. Applied egg-rr 22.3%

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

       1. associate--l+ 24.0%

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

       2. unpow2 24.0%

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

       3. sqr-neg 24.0%

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

       4. rem-square-sqrt 26.0%

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

       5. sub-neg 26.0%

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

       6. remove-double-neg 26.0%

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

   14. Simplified 26.0%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{x} + \sqrt{y + 1}\right)\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.2e+22)
   (-
    (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ 1.0 (+ (sqrt x) (sqrt (+ y 1.0)))))
    (sqrt y))
   (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (1.0 + (sqrt(x) + sqrt((y + 1.0))))) - sqrt(y);
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.2d+22) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (1.0d0 + (sqrt(x) + sqrt((y + 1.0d0))))) - sqrt(y)
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (1.0 + (Math.sqrt(x) + Math.sqrt((y + 1.0))))) - Math.sqrt(y);
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.2e+22:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (1.0 + (math.sqrt(x) + math.sqrt((y + 1.0))))) - math.sqrt(y)
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.2e+22)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(1.0 + Float64(sqrt(x) + sqrt(Float64(y + 1.0))))) - sqrt(y));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.2e+22)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (1.0 + (sqrt(x) + sqrt((y + 1.0))))) - sqrt(y);
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.2e+22], N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e22

   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. +-commutative 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 49.1%

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

   6. Taylor expanded in t around inf 27.4%

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

      1. +-commutative 27.4%

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

      2. associate-+r- 27.4%

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

      3. associate-+r- 27.2%

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

      4. sub-neg 27.2%

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

      5. associate-+l+ 27.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 30.1%

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

      8. sqr-neg 30.1%

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

      9. add-sqr-sqrt 30.1%

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

   8. Applied egg-rr 30.1%

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

   if 1.2e22 < y

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-+r- 44.4%

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

      4. associate-+l- 23.6%

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

      5. associate-+r- 4.3%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 22.8%

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

      2. +-commutative 22.8%

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

      3. +-commutative 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around inf 22.2%

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

      1. mul-1-neg 22.2%

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

   10. Simplified 22.2%

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

       1. flip-+ 22.2%

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

       2. add-sqr-sqrt 22.3%

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

       3. pow2 22.3%

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

   12. Applied egg-rr 22.3%

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

       1. associate--l+ 24.0%

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

       2. unpow2 24.0%

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

       3. sqr-neg 24.0%

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

       4. rem-square-sqrt 26.0%

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

       5. sub-neg 26.0%

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

       6. remove-double-neg 26.0%

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

   14. Simplified 26.0%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{x} + \sqrt{y + 1}\right)\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.2e+22)
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
   (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.2d+22) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+22) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.2e+22:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.2e+22)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.2e+22)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.2e+22], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e22

   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. +-commutative 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 49.1%

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

   6. Taylor expanded in t around inf 27.4%

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

   if 1.2e22 < y

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-+r- 44.4%

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

      4. associate-+l- 23.6%

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

      5. associate-+r- 4.3%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in t around inf 3.6%

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

      1. associate--l+ 22.8%

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

      2. +-commutative 22.8%

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

      3. +-commutative 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around inf 22.2%

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

      1. mul-1-neg 22.2%

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

   10. Simplified 22.2%

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

       1. flip-+ 22.2%

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

       2. add-sqr-sqrt 22.3%

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

       3. pow2 22.3%

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

   12. Applied egg-rr 22.3%

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

       1. associate--l+ 24.0%

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

       2. unpow2 24.0%

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

       3. sqr-neg 24.0%

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

       4. rem-square-sqrt 26.0%

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

       5. sub-neg 26.0%

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

       6. remove-double-neg 26.0%

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

   14. Simplified 26.0%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.32e-21)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (- 2.0 (sqrt x)) (sqrt y)))
   (if (<= y 3.55e+19)
     (+
      1.0
      (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x))))
     (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.32e-21) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	} else if (y <= 3.55e+19) {
		tmp = 1.0 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.32d-21) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((2.0d0 - sqrt(x)) - sqrt(y))
    else if (y <= 3.55d+19) then
        tmp = 1.0d0 + ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.32e-21) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(x)) - Math.sqrt(y));
	} else if (y <= 3.55e+19) {
		tmp = 1.0 + ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.32e-21:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((2.0 - math.sqrt(x)) - math.sqrt(y))
	elif y <= 3.55e+19:
		tmp = 1.0 + ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.32e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(x)) - sqrt(y)));
	elseif (y <= 3.55e+19)
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.32e-21)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	elseif (y <= 3.55e+19)
		tmp = 1.0 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.32e-21], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.55e+19], N[(1.0 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.32e-21

   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. +-commutative 97.7%

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

      3. +-commutative 97.7%

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

      4. +-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 x around 0 49.0%

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

   6. Taylor expanded in t around inf 27.4%

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

   7. Taylor expanded in y around 0 27.4%

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

      1. associate--r+ 27.4%

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

   9. Simplified 27.4%

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

   if 1.32e-21 < y < 3.55e19

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 96.7%

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

      4. associate-+l- 96.7%

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

      5. associate-+r- 96.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in t around inf 14.5%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

      3. +-commutative 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around 0 14.0%

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

   9. Taylor expanded in z around inf 13.0%

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

       1. associate--l+ 13.0%

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

       2. +-commutative 13.0%

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

   11. Simplified 13.0%

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

   if 3.55e19 < y

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 44.6%

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

      4. associate-+l- 24.0%

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

      5. associate-+r- 4.7%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 23.1%

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

      2. +-commutative 23.1%

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

      3. +-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   10. Simplified 22.5%

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

       1. flip-+ 22.5%

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

       2. add-sqr-sqrt 22.5%

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

       3. pow2 22.5%

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

   12. Applied egg-rr 22.5%

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

       1. associate--l+ 24.2%

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

       2. unpow2 24.2%

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

       3. sqr-neg 24.2%

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

       4. rem-square-sqrt 26.3%

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

       5. sub-neg 26.3%

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

       6. remove-double-neg 26.3%

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

   14. Simplified 26.3%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.32e-21)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (- 2.0 (sqrt x)) (sqrt y)))
   (if (<= y 5e+19)
     (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))
     (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.32e-21) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	} else if (y <= 5e+19) {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.32d-21) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((2.0d0 - sqrt(x)) - sqrt(y))
    else if (y <= 5d+19) then
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.32e-21) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(x)) - Math.sqrt(y));
	} else if (y <= 5e+19) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.32e-21:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((2.0 - math.sqrt(x)) - math.sqrt(y))
	elif y <= 5e+19:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.32e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(x)) - sqrt(y)));
	elseif (y <= 5e+19)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.32e-21)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	elseif (y <= 5e+19)
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.32e-21], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+19], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.32e-21

   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. +-commutative 97.7%

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

      3. +-commutative 97.7%

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

      4. +-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 x around 0 49.0%

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

   6. Taylor expanded in t around inf 27.4%

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

   7. Taylor expanded in y around 0 27.4%

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

      1. associate--r+ 27.4%

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

   9. Simplified 27.4%

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

   if 1.32e-21 < y < 5e19

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 96.7%

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

      4. associate-+l- 96.7%

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

      5. associate-+r- 96.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in t around inf 14.5%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

      3. +-commutative 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around 0 14.0%

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

   9. Taylor expanded in z around inf 17.2%

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

       1. associate--l+ 17.2%

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

   11. Simplified 17.2%

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

   if 5e19 < y

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 44.6%

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

      4. associate-+l- 24.0%

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

      5. associate-+r- 4.7%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 23.1%

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

      2. +-commutative 23.1%

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

      3. +-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   10. Simplified 22.5%

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

       1. flip-+ 22.5%

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

       2. add-sqr-sqrt 22.5%

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

       3. pow2 22.5%

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

   12. Applied egg-rr 22.5%

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

       1. associate--l+ 24.2%

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

       2. unpow2 24.2%

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

       3. sqr-neg 24.2%

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

       4. rem-square-sqrt 26.3%

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

       5. sub-neg 26.3%

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

       6. remove-double-neg 26.3%

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

   14. Simplified 26.3%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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 7.7e-25)
   (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (+ (sqrt x) (sqrt z)))))
   (if (<= y 5e+19)
     (- (+ 1.0 (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x)))
     (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.7e-25) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	} else if (y <= 5e+19) {
		tmp = (1.0 + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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 <= 7.7d-25) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + (sqrt(x) + sqrt(z))))
    else if (y <= 5d+19) then
        tmp = (1.0d0 + sqrt((y + 1.0d0))) - (sqrt(y) + sqrt(x))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + 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 <= 7.7e-25) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + (Math.sqrt(x) + Math.sqrt(z))));
	} else if (y <= 5e+19) {
		tmp = (1.0 + Math.sqrt((y + 1.0))) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + 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 <= 7.7e-25:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + (math.sqrt(x) + math.sqrt(z))))
	elif y <= 5e+19:
		tmp = (1.0 + math.sqrt((y + 1.0))) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + 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 <= 7.7e-25)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + Float64(sqrt(x) + sqrt(z)))));
	elseif (y <= 5e+19)
		tmp = Float64(Float64(1.0 + sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + 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 <= 7.7e-25)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + (sqrt(x) + sqrt(z))));
	elseif (y <= 5e+19)
		tmp = (1.0 + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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, 7.7e-25], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+19], N[(N[(1.0 + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.7000000000000002e-25

   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. +-commutative 97.7%

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

      2. associate-+r+ 97.6%

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

      3. associate-+r- 97.7%

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

      4. associate-+l- 97.7%

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

      5. associate-+r- 97.7%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in t around inf 15.2%

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

      1. associate--l+ 19.5%

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

      2. +-commutative 19.5%

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

      3. +-commutative 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in x around 0 13.2%

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

   9. Taylor expanded in y around 0 13.2%

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

       1. associate--l+ 32.1%

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

       2. +-commutative 32.1%

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

       3. associate-+r+ 32.1%

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

       4. +-commutative 32.1%

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

   11. Simplified 32.1%

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

   if 7.7000000000000002e-25 < y < 5e19

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 96.7%

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

      4. associate-+l- 96.7%

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

      5. associate-+r- 96.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in t around inf 14.5%

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

      1. associate--l+ 18.8%

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

      2. +-commutative 18.8%

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

      3. +-commutative 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around 0 14.0%

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

   9. Taylor expanded in z around inf 17.2%

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

   if 5e19 < y

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 44.6%

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

      4. associate-+l- 24.0%

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

      5. associate-+r- 4.7%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 23.1%

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

      2. +-commutative 23.1%

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

      3. +-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   10. Simplified 22.5%

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

       1. flip-+ 22.5%

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

       2. add-sqr-sqrt 22.5%

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

       3. pow2 22.5%

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

   12. Applied egg-rr 22.5%

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

       1. associate--l+ 24.2%

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

       2. unpow2 24.2%

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

       3. sqr-neg 24.2%

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

       4. rem-square-sqrt 26.3%

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

       5. sub-neg 26.3%

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

       6. remove-double-neg 26.3%

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

   14. Simplified 26.3%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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+19)
   (- (+ 1.0 (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x)))
   (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (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+19) {
		tmp = (1.0 + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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+19) then
        tmp = (1.0d0 + sqrt((y + 1.0d0))) - (sqrt(y) + sqrt(x))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + 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+19) {
		tmp = (1.0 + Math.sqrt((y + 1.0))) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + 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+19:
		tmp = (1.0 + math.sqrt((y + 1.0))) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + 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+19)
		tmp = Float64(Float64(1.0 + sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + 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+19)
		tmp = (1.0 + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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+19], N[(N[(1.0 + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e19

   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. +-commutative 97.6%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.6%

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

      4. associate-+l- 97.6%

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

      5. associate-+r- 97.6%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 19.4%

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

      2. +-commutative 19.4%

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

      3. +-commutative 19.4%

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

   7. Simplified 19.4%

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

   8. Taylor expanded in x around 0 13.2%

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

   9. Taylor expanded in z around inf 20.8%

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

   if 3.6e19 < y

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 44.6%

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

      4. associate-+l- 24.0%

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

      5. associate-+r- 4.7%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 23.1%

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

      2. +-commutative 23.1%

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

      3. +-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   10. Simplified 22.5%

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

       1. flip-+ 22.5%

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

       2. add-sqr-sqrt 22.5%

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

       3. pow2 22.5%

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

   12. Applied egg-rr 22.5%

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

       1. associate--l+ 24.2%

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

       2. unpow2 24.2%

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

       3. sqr-neg 24.2%

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

       4. rem-square-sqrt 26.3%

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

       5. sub-neg 26.3%

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

       6. remove-double-neg 26.3%

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

   14. Simplified 26.3%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

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+19)
   (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))
   (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (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+19) {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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+19) then
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + 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+19) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + 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+19:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt((1.0 + x)) + 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+19)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + 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+19)
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt((1.0 + x)) + 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+19], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e19

   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. +-commutative 97.6%

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

      2. associate-+r+ 97.5%

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

      3. associate-+r- 97.6%

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

      4. associate-+l- 97.6%

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

      5. associate-+r- 97.6%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 19.4%

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

      2. +-commutative 19.4%

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

      3. +-commutative 19.4%

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

   7. Simplified 19.4%

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

   8. Taylor expanded in x around 0 13.2%

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

   9. Taylor expanded in z around inf 20.8%

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

       1. associate--l+ 20.8%

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

   11. Simplified 20.8%

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

   if 3.6e19 < y

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 44.6%

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

      4. associate-+l- 24.0%

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

      5. associate-+r- 4.7%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 23.1%

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

      2. +-commutative 23.1%

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

      3. +-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   10. Simplified 22.5%

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

       1. flip-+ 22.5%

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

       2. add-sqr-sqrt 22.5%

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

       3. pow2 22.5%

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

   12. Applied egg-rr 22.5%

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

       1. associate--l+ 24.2%

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

       2. unpow2 24.2%

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

       3. sqr-neg 24.2%

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

       4. rem-square-sqrt 26.3%

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

       5. sub-neg 26.3%

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

       6. remove-double-neg 26.3%

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

   14. Simplified 26.3%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. +-commutative 92.3%

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

   2. associate-+r+ 92.3%

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

   3. associate-+r- 71.9%

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

   4. associate-+l- 61.9%

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

   5. associate-+r- 52.6%

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

3. Simplified 42.2%

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

5. Taylor expanded in t around inf 9.7%

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

   1. associate--l+ 21.2%

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

   2. +-commutative 21.2%

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

   3. +-commutative 21.2%

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

7. Simplified 21.2%

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

8. Taylor expanded in x around inf 16.4%

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

   1. mul-1-neg 16.4%

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

10. Simplified 16.4%

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

    1. flip-+ 16.4%

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

    2. add-sqr-sqrt 16.6%

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

    3. pow2 16.6%

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

12. Applied egg-rr 16.6%

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

    1. associate--l+ 17.8%

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

    2. unpow2 17.8%

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

    3. sqr-neg 17.8%

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

    4. rem-square-sqrt 19.1%

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

    5. sub-neg 19.1%

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

    6. remove-double-neg 19.1%

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

14. Simplified 19.1%

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

Alternative 16: 39.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 60000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 60000000.0) (- (sqrt (+ 1.0 x)) (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 <= 60000000.0) {
		tmp = sqrt((1.0 + x)) - 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 <= 60000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - 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 <= 60000000.0) {
		tmp = Math.sqrt((1.0 + x)) - 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 <= 60000000.0:
		tmp = math.sqrt((1.0 + x)) - 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 <= 60000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - 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 <= 60000000.0)
		tmp = sqrt((1.0 + x)) - 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, 60000000.0], N[(N[Sqrt[N[(1.0 + x), $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 < 6e7

   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. +-commutative 97.3%

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

      2. associate-+r+ 97.3%

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

      3. associate-+r- 58.4%

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

      4. associate-+l- 55.9%

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

      5. associate-+r- 52.3%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 35.7%

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

      2. +-commutative 35.7%

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

      3. +-commutative 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in x around inf 28.3%

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

      1. mul-1-neg 28.3%

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

   10. Simplified 28.3%

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

       1. unsub-neg 28.3%

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

   12. Applied egg-rr 28.3%

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

   if 6e7 < x

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. associate-+r- 86.7%

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

      4. associate-+l- 68.5%

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

      5. associate-+r- 52.9%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in t around inf 3.7%

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

      1. associate--l+ 5.3%

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

      2. +-commutative 5.3%

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

      3. +-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in x around inf 3.4%

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

      1. mul-1-neg 3.4%

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

   10. Simplified 3.4%

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

   11. Taylor expanded in x around inf 9.1%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 39.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 3.7)
   (- (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))) (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 <= 3.7) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - 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 <= 3.7d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))) - 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 <= 3.7) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - 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 <= 3.7:
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - 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 <= 3.7)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))) - 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 <= 3.7)
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - 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, 3.7], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[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 < 3.7000000000000002

   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. +-commutative 97.7%

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

      2. associate-+r+ 97.7%

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

      3. associate-+r- 58.9%

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

      4. associate-+l- 56.3%

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

      5. associate-+r- 52.7%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in t around inf 14.8%

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

      1. associate--l+ 35.6%

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

      2. +-commutative 35.6%

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

      3. +-commutative 35.6%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in x around inf 28.2%

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

      1. mul-1-neg 28.2%

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

   10. Simplified 28.2%

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

   11. Taylor expanded in x around 0 28.2%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) - \sqrt{x}} \]

   if 3.7000000000000002 < x

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. associate-+r+ 86.7%

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

      3. associate-+r- 85.6%

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

      4. associate-+l- 67.8%

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

      5. associate-+r- 52.4%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 6.1%

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

      2. +-commutative 6.1%

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

      3. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around inf 4.0%

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

      1. mul-1-neg 4.0%

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

   10. Simplified 4.0%

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

   11. Taylor expanded in x around inf 9.4%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 1.1)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (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 <= 1.1) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - 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 <= 1.1d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - 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 <= 1.1) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - 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 <= 1.1:
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - 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 <= 1.1)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - 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 <= 1.1)
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - 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, 1.1], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   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. +-commutative 97.7%

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

      2. associate-+r+ 97.7%

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

      3. associate-+r- 58.5%

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

      4. associate-+l- 56.0%

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

      5. associate-+r- 52.4%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in t around inf 14.7%

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

      1. associate--l+ 35.7%

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

      2. +-commutative 35.7%

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

      3. +-commutative 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in x around inf 28.3%

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

      1. mul-1-neg 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in x around 0 28.3%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ 28.3%

          \[\leadsto \color{blue}{1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)} \]

   13. Simplified 28.3%

       \[\leadsto \color{blue}{1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. associate-+r- 85.7%

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

      4. associate-+l- 68.0%

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

      5. associate-+r- 52.8%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around inf 4.5%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

      3. +-commutative 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in x around 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 9.5%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 39.3% accurate, 7.3× speedup

Math

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

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) (+ 1.0 (- (* x 0.5) (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 <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - 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 <= 1.0d0) then
        tmp = 1.0d0 + ((x * 0.5d0) - 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 <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - 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 <= 1.0:
		tmp = 1.0 + ((x * 0.5) - 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 <= 1.0)
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) - 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 <= 1.0)
		tmp = 1.0 + ((x * 0.5) - 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, 1.0], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. +-commutative 97.7%

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

      2. associate-+r+ 97.7%

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

      3. associate-+r- 58.5%

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

      4. associate-+l- 56.0%

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

      5. associate-+r- 52.4%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in t around inf 14.7%

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

      1. associate--l+ 35.7%

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

      2. +-commutative 35.7%

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

      3. +-commutative 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in x around inf 28.3%

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

      1. mul-1-neg 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in x around 0 28.3%

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

       1. associate--l+ 28.3%

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

   13. Simplified 28.3%

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

   if 1 < x

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. associate-+r- 85.7%

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

      4. associate-+l- 68.0%

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

      5. associate-+r- 52.8%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around inf 4.5%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

      3. +-commutative 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in x around 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 9.5%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

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

Alternative 20: 39.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.049:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

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 0.049) (- 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 <= 0.049) {
		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 <= 0.049d0) 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 <= 0.049) {
		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 <= 0.049:
		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 <= 0.049)
		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 <= 0.049)
		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, 0.049], 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 < 0.049000000000000002

   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. +-commutative 97.7%

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

      2. associate-+r+ 97.7%

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

      3. associate-+r- 58.5%

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

      4. associate-+l- 56.0%

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

      5. associate-+r- 52.4%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in t around inf 14.7%

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

      1. associate--l+ 35.7%

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

      2. +-commutative 35.7%

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

      3. +-commutative 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in x around inf 28.3%

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

      1. mul-1-neg 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in x around 0 28.3%

       \[\leadsto \color{blue}{1 - \sqrt{x}} \]

   if 0.049000000000000002 < x

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. associate-+r- 85.7%

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

      4. associate-+l- 68.0%

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

      5. associate-+r- 52.8%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around inf 4.5%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

      3. +-commutative 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in x around 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 9.5%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. +-commutative 92.3%

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

   2. associate-+r+ 92.3%

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

   3. associate-+r- 71.9%

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

   4. associate-+l- 61.9%

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

   5. associate-+r- 52.6%

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

3. Simplified 42.2%

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

5. Taylor expanded in t around inf 9.7%

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

   1. associate--l+ 21.2%

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

   2. +-commutative 21.2%

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

   3. +-commutative 21.2%

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

7. Simplified 21.2%

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

8. Taylor expanded in x around inf 16.4%

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

   1. mul-1-neg 16.4%

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

10. Simplified 16.4%

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

11. Taylor expanded in x around 0 15.0%

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

Alternative 22: 2.0% 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 92.3%

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

   1. +-commutative 92.3%

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

   2. associate-+r+ 92.3%

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

   3. associate-+r- 71.9%

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

   4. associate-+l- 61.9%

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

   5. associate-+r- 52.6%

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

3. Simplified 42.2%

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

5. Taylor expanded in t around inf 9.7%

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

   1. associate--l+ 21.2%

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

   2. +-commutative 21.2%

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

   3. +-commutative 21.2%

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

7. Simplified 21.2%

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

8. Taylor expanded in x around 0 8.5%

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

9. Taylor expanded in x around inf 1.6%

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

    1. neg-mul-1 1.6%

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

11. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]
12. Add Preprocessing

Developer target: 99.5% 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 2024087 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

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

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))