Main:z from

Percentage Accurate: 91.5% → 99.4%

Time: 1.1min

Alternatives: 19

Speedup: 1.9×

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: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \frac{1}{\sqrt{y + 1} + \sqrt{y}}\\ t_4 := t\_2 - \sqrt{x}\\ t_5 := \sqrt{z + 1}\\ \mathbf{if}\;t\_4 \leq 0.02:\\ \;\;\;\;\left(\left(t\_1 - \left(\sqrt{t} - \left(t\_5 - \sqrt{z}\right)\right)\right) + t\_3\right) + \frac{1}{\sqrt{x} + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_5 + \sqrt{z}} + \left(t\_1 - \sqrt{t}\right)\right) + \left(t\_4 + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ t 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
        (t_4 (- t_2 (sqrt x)))
        (t_5 (sqrt (+ z 1.0))))
   (if (<= t_4 0.02)
     (+ (+ (- t_1 (- (sqrt t) (- t_5 (sqrt z)))) t_3) (/ 1.0 (+ (sqrt x) t_2)))
     (+ (+ (/ 1.0 (+ t_5 (sqrt z))) (- t_1 (sqrt t))) (+ t_4 t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = 1.0 / (sqrt((y + 1.0)) + sqrt(y));
	double t_4 = t_2 - sqrt(x);
	double t_5 = sqrt((z + 1.0));
	double tmp;
	if (t_4 <= 0.02) {
		tmp = ((t_1 - (sqrt(t) - (t_5 - sqrt(z)))) + t_3) + (1.0 / (sqrt(x) + t_2));
	} else {
		tmp = ((1.0 / (t_5 + sqrt(z))) + (t_1 - sqrt(t))) + (t_4 + t_3);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    t_3 = 1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))
    t_4 = t_2 - sqrt(x)
    t_5 = sqrt((z + 1.0d0))
    if (t_4 <= 0.02d0) then
        tmp = ((t_1 - (sqrt(t) - (t_5 - sqrt(z)))) + t_3) + (1.0d0 / (sqrt(x) + t_2))
    else
        tmp = ((1.0d0 / (t_5 + sqrt(z))) + (t_1 - sqrt(t))) + (t_4 + t_3)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = 1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y));
	double t_4 = t_2 - Math.sqrt(x);
	double t_5 = Math.sqrt((z + 1.0));
	double tmp;
	if (t_4 <= 0.02) {
		tmp = ((t_1 - (Math.sqrt(t) - (t_5 - Math.sqrt(z)))) + t_3) + (1.0 / (Math.sqrt(x) + t_2));
	} else {
		tmp = ((1.0 / (t_5 + Math.sqrt(z))) + (t_1 - Math.sqrt(t))) + (t_4 + t_3);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0))
	t_2 = math.sqrt((x + 1.0))
	t_3 = 1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))
	t_4 = t_2 - math.sqrt(x)
	t_5 = math.sqrt((z + 1.0))
	tmp = 0
	if t_4 <= 0.02:
		tmp = ((t_1 - (math.sqrt(t) - (t_5 - math.sqrt(z)))) + t_3) + (1.0 / (math.sqrt(x) + t_2))
	else:
		tmp = ((1.0 / (t_5 + math.sqrt(z))) + (t_1 - math.sqrt(t))) + (t_4 + t_3)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))
	t_4 = Float64(t_2 - sqrt(x))
	t_5 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t_4 <= 0.02)
		tmp = Float64(Float64(Float64(t_1 - Float64(sqrt(t) - Float64(t_5 - sqrt(z)))) + t_3) + Float64(1.0 / Float64(sqrt(x) + t_2)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_5 + sqrt(z))) + Float64(t_1 - sqrt(t))) + Float64(t_4 + t_3));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0));
	t_2 = sqrt((x + 1.0));
	t_3 = 1.0 / (sqrt((y + 1.0)) + sqrt(y));
	t_4 = t_2 - sqrt(x);
	t_5 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t_4 <= 0.02)
		tmp = ((t_1 - (sqrt(t) - (t_5 - sqrt(z)))) + t_3) + (1.0 / (sqrt(x) + t_2));
	else
		tmp = ((1.0 / (t_5 + sqrt(z))) + (t_1 - sqrt(t))) + (t_4 + t_3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.02], N[(N[(N[(t$95$1 - N[(N[Sqrt[t], $MachinePrecision] - N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$5 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.0200000000000000004

   1. Initial program 80.2%

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

      1. associate-+l+ 80.2%

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

      2. associate-+l+ 80.2%

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

      3. +-commutative 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-+l- 66.2%

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

      6. +-commutative 66.2%

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

      7. +-commutative 66.2%

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

   3. Simplified 66.2%

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

      1. flip-- 66.2%

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

      2. add-sqr-sqrt 36.9%

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

      3. +-commutative 36.9%

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

      4. add-sqr-sqrt 66.8%

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

      5. +-commutative 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. associate--l+ 71.4%

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

      2. +-inverses 71.4%

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

      3. metadata-eval 71.4%

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

   8. Simplified 71.4%

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

      1. flip-- 71.9%

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

      2. add-sqr-sqrt 58.6%

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

      3. add-sqr-sqrt 72.0%

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

   10. Applied egg-rr 72.0%

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

       1. associate--l+ 75.0%

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

       2. +-inverses 75.0%

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

       3. metadata-eval 75.0%

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

   12. Simplified 75.0%

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

   if 0.0200000000000000004 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. sub-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. +-commutative 96.4%

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

   3. Simplified 96.4%

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

      1. flip-- 96.5%

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

      2. add-sqr-sqrt 72.4%

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

      3. add-sqr-sqrt 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. associate--l+ 97.1%

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

      2. +-inverses 97.1%

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

      3. metadata-eval 97.1%

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

   8. Simplified 97.1%

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

      1. flip-- 82.0%

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

      2. add-sqr-sqrt 58.8%

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

      3. add-sqr-sqrt 82.0%

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

   10. Applied egg-rr 97.3%

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

       1. associate--l+ 82.4%

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

       2. +-inverses 82.4%

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

       3. metadata-eval 82.4%

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

   12. Simplified 98.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.02:\\ \;\;\;\;\left(\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := \sqrt{z + 1}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{x + 1}\\ t_6 := t\_5 - \sqrt{x}\\ t_7 := t\_4 + \left(t\_6 + t\_2\right)\\ \mathbf{if}\;t\_7 \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_5}\\ \mathbf{elif}\;t\_7 \leq 2.5:\\ \;\;\;\;t\_6 + \left(\frac{1}{t\_3 + \sqrt{z}} + \frac{1}{t\_1 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + 1\right) + \left(t\_4 + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ x 1.0)))
        (t_6 (- t_5 (sqrt x)))
        (t_7 (+ t_4 (+ t_6 t_2))))
   (if (<= t_7 1e-6)
     (/ 1.0 (+ (sqrt x) t_5))
     (if (<= t_7 2.5)
       (+ t_6 (+ (/ 1.0 (+ t_3 (sqrt z))) (/ 1.0 (+ t_1 (sqrt y)))))
       (+ (+ t_2 1.0) (+ t_4 (/ 1.0 (+ (sqrt (+ t 1.0)) (sqrt t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((x + 1.0));
	double t_6 = t_5 - sqrt(x);
	double t_7 = t_4 + (t_6 + t_2);
	double tmp;
	if (t_7 <= 1e-6) {
		tmp = 1.0 / (sqrt(x) + t_5);
	} else if (t_7 <= 2.5) {
		tmp = t_6 + ((1.0 / (t_3 + sqrt(z))) + (1.0 / (t_1 + sqrt(y))));
	} else {
		tmp = (t_2 + 1.0) + (t_4 + (1.0 / (sqrt((t + 1.0)) + sqrt(t))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((z + 1.0d0))
    t_4 = t_3 - sqrt(z)
    t_5 = sqrt((x + 1.0d0))
    t_6 = t_5 - sqrt(x)
    t_7 = t_4 + (t_6 + t_2)
    if (t_7 <= 1d-6) then
        tmp = 1.0d0 / (sqrt(x) + t_5)
    else if (t_7 <= 2.5d0) then
        tmp = t_6 + ((1.0d0 / (t_3 + sqrt(z))) + (1.0d0 / (t_1 + sqrt(y))))
    else
        tmp = (t_2 + 1.0d0) + (t_4 + (1.0d0 / (sqrt((t + 1.0d0)) + sqrt(t))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = t_3 - Math.sqrt(z);
	double t_5 = Math.sqrt((x + 1.0));
	double t_6 = t_5 - Math.sqrt(x);
	double t_7 = t_4 + (t_6 + t_2);
	double tmp;
	if (t_7 <= 1e-6) {
		tmp = 1.0 / (Math.sqrt(x) + t_5);
	} else if (t_7 <= 2.5) {
		tmp = t_6 + ((1.0 / (t_3 + Math.sqrt(z))) + (1.0 / (t_1 + Math.sqrt(y))));
	} else {
		tmp = (t_2 + 1.0) + (t_4 + (1.0 / (Math.sqrt((t + 1.0)) + Math.sqrt(t))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((z + 1.0))
	t_4 = t_3 - math.sqrt(z)
	t_5 = math.sqrt((x + 1.0))
	t_6 = t_5 - math.sqrt(x)
	t_7 = t_4 + (t_6 + t_2)
	tmp = 0
	if t_7 <= 1e-6:
		tmp = 1.0 / (math.sqrt(x) + t_5)
	elif t_7 <= 2.5:
		tmp = t_6 + ((1.0 / (t_3 + math.sqrt(z))) + (1.0 / (t_1 + math.sqrt(y))))
	else:
		tmp = (t_2 + 1.0) + (t_4 + (1.0 / (math.sqrt((t + 1.0)) + math.sqrt(t))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(x + 1.0))
	t_6 = Float64(t_5 - sqrt(x))
	t_7 = Float64(t_4 + Float64(t_6 + t_2))
	tmp = 0.0
	if (t_7 <= 1e-6)
		tmp = Float64(1.0 / Float64(sqrt(x) + t_5));
	elseif (t_7 <= 2.5)
		tmp = Float64(t_6 + Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + Float64(1.0 / Float64(t_1 + sqrt(y)))));
	else
		tmp = Float64(Float64(t_2 + 1.0) + Float64(t_4 + Float64(1.0 / Float64(sqrt(Float64(t + 1.0)) + sqrt(t)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((z + 1.0));
	t_4 = t_3 - sqrt(z);
	t_5 = sqrt((x + 1.0));
	t_6 = t_5 - sqrt(x);
	t_7 = t_4 + (t_6 + t_2);
	tmp = 0.0;
	if (t_7 <= 1e-6)
		tmp = 1.0 / (sqrt(x) + t_5);
	elseif (t_7 <= 2.5)
		tmp = t_6 + ((1.0 / (t_3 + sqrt(z))) + (1.0 / (t_1 + sqrt(y))));
	else
		tmp = (t_2 + 1.0) + (t_4 + (1.0 / (sqrt((t + 1.0)) + sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$4 + N[(t$95$6 + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 1e-6], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.5], N[(t$95$6 + N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 1.0), $MachinePrecision] + N[(t$95$4 + N[(1.0 / N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 9.99999999999999955e-7

   1. Initial program 44.1%

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

      1. +-commutative 44.1%

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

      2. associate-+r+ 44.1%

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

      3. associate-+r- 44.1%

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

      4. associate-+l- 9.9%

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

      5. associate-+r- 9.9%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 5.6%

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

      2. +-commutative 5.6%

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

      3. associate--l+ 5.3%

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

      4. +-commutative 5.3%

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

      5. associate-+r+ 5.2%

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

   7. Simplified 5.2%

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

   8. Taylor expanded in z around inf 4.8%

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

      1. +-commutative 4.8%

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

   10. Simplified 4.8%

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

   11. Taylor expanded in y around inf 3.3%

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

       1. flip-- 3.3%

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

       2. add-sqr-sqrt 3.9%

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

       3. add-sqr-sqrt 3.3%

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

       4. +-commutative 3.3%

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

       5. +-commutative 3.3%

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

       6. +-commutative 3.3%

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

   13. Applied egg-rr 3.3%

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

       1. associate--l+ 19.3%

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

       2. +-inverses 19.3%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 19.3%

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

   15. Simplified 19.3%

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

   if 9.99999999999999955e-7 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 2.5

   1. Initial program 95.4%

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

      1. associate-+l+ 95.4%

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

      2. sub-neg 95.4%

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

      3. sub-neg 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

      6. +-commutative 95.4%

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

   3. Simplified 95.4%

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

      1. flip-- 95.8%

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

      2. add-sqr-sqrt 74.5%

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

      3. add-sqr-sqrt 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. associate--l+ 96.7%

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

      2. +-inverses 96.7%

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

      3. metadata-eval 96.7%

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

   8. Simplified 96.7%

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

      1. flip-- 81.5%

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

      2. add-sqr-sqrt 63.9%

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

      3. add-sqr-sqrt 81.6%

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

   10. Applied egg-rr 96.9%

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

       1. associate--l+ 81.9%

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

       2. +-inverses 81.9%

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

       3. metadata-eval 81.9%

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

   12. Simplified 97.7%

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

   13. Taylor expanded in t around inf 29.5%

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

       1. sub-neg 29.5%

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

       2. +-commutative 29.5%

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

       3. +-commutative 29.5%

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

       4. associate-+l+ 53.0%

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

       5. +-commutative 53.0%

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

       6. sub-neg 53.0%

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

   15. Simplified 53.0%

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

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)))

   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. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 93.9%

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

      1. flip-- 93.8%

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

      2. add-sqr-sqrt 75.7%

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

      3. +-commutative 75.7%

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

      4. add-sqr-sqrt 96.0%

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

      5. +-commutative 96.0%

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

   7. Applied egg-rr 96.0%

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

      1. +-commutative 96.0%

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

      2. associate--l+ 96.5%

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

      3. +-inverses 96.5%

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

      4. metadata-eval 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

   9. Simplified 96.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 - \sqrt{x}\\ \mathbf{if}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(t\_2 + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (- t_1 (sqrt x))))
   (if (<= t_2 1e-6)
     (/ 1.0 (+ (sqrt x) t_1))
     (+
      (+ (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t)))
      (+ t_2 (/ 1.0 (+ (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((x + 1.0));
	double t_2 = t_1 - sqrt(x);
	double tmp;
	if (t_2 <= 1e-6) {
		tmp = 1.0 / (sqrt(x) + t_1);
	} else {
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) + (t_2 + (1.0 / (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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = t_1 - sqrt(x)
    if (t_2 <= 1d-6) then
        tmp = 1.0d0 / (sqrt(x) + t_1)
    else
        tmp = ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) + (t_2 + (1.0d0 / (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((x + 1.0));
	double t_2 = t_1 - Math.sqrt(x);
	double tmp;
	if (t_2 <= 1e-6) {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	} else {
		tmp = ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) + (t_2 + (1.0 / (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((x + 1.0))
	t_2 = t_1 - math.sqrt(x)
	tmp = 0
	if t_2 <= 1e-6:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	else:
		tmp = ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) + (t_2 + (1.0 / (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(x + 1.0))
	t_2 = Float64(t_1 - sqrt(x))
	tmp = 0.0
	if (t_2 <= 1e-6)
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) + Float64(t_2 + Float64(1.0 / 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((x + 1.0));
	t_2 = t_1 - sqrt(x);
	tmp = 0.0;
	if (t_2 <= 1e-6)
		tmp = 1.0 / (sqrt(x) + t_1);
	else
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) + (t_2 + (1.0 / (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-6], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / 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] + N[(t$95$2 + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 9.99999999999999955e-7

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. associate-+r+ 80.1%

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

      3. associate-+r- 80.1%

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

      4. associate-+l- 60.6%

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

      5. associate-+r- 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around inf 3.3%

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

      1. associate--l+ 4.5%

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

      2. +-commutative 4.5%

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

      3. associate--l+ 4.4%

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

      4. +-commutative 4.4%

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

      5. associate-+r+ 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in z around inf 4.1%

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

      1. +-commutative 4.1%

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

   10. Simplified 4.1%

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

   11. Taylor expanded in y around inf 3.2%

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

       1. flip-- 3.2%

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

       2. add-sqr-sqrt 3.9%

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

       3. add-sqr-sqrt 3.2%

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

       4. +-commutative 3.2%

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

       5. +-commutative 3.2%

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

       6. +-commutative 3.2%

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

   13. Applied egg-rr 3.2%

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

       1. associate--l+ 10.2%

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

       2. +-inverses 10.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 10.2%

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

   15. Simplified 10.2%

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

   if 9.99999999999999955e-7 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. sub-neg 95.7%

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

      3. sub-neg 95.7%

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

      4. +-commutative 95.7%

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

      5. +-commutative 95.7%

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

      6. +-commutative 95.7%

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

   3. Simplified 95.7%

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

      1. flip-- 95.8%

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

      2. add-sqr-sqrt 73.1%

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

      3. add-sqr-sqrt 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. associate--l+ 96.4%

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

      2. +-inverses 96.4%

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

      3. metadata-eval 96.4%

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

   8. Simplified 96.4%

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

      1. flip-- 80.4%

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

      2. add-sqr-sqrt 58.4%

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

      3. add-sqr-sqrt 80.4%

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

   10. Applied egg-rr 96.6%

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

       1. associate--l+ 80.7%

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

       2. +-inverses 80.7%

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

       3. metadata-eval 80.7%

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

   12. Simplified 97.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;z \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + t\_1\right) + \left(t\_2 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(t\_1 + t\_2\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_2 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= z 5e+42)
     (+ (+ (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))) t_1) (+ t_2 1.0))
     (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (+ t_1 t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (z <= 5e+42) {
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_1) + (t_2 + 1.0);
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (t_1 + 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((t + 1.0d0)) - sqrt(t)
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    if (z <= 5d+42) then
        tmp = ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + t_1) + (t_2 + 1.0d0)
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (t_1 + 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((t + 1.0)) - Math.sqrt(t);
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (z <= 5e+42) {
		tmp = ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + t_1) + (t_2 + 1.0);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (t_1 + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if z <= 5e+42:
		tmp = ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + t_1) + (t_2 + 1.0)
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (t_1 + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (z <= 5e+42)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + t_1) + Float64(t_2 + 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(t_1 + 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((t + 1.0)) - sqrt(t);
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (z <= 5e+42)
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_1) + (t_2 + 1.0);
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (t_1 + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5e+42], N[(N[(N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000007e42

   1. Initial program 91.1%

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

      1. associate-+l+ 91.1%

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

      2. sub-neg 91.1%

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

      3. sub-neg 91.1%

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

      4. +-commutative 91.1%

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

      5. +-commutative 91.1%

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

      6. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 51.4%

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

      1. flip-- 91.7%

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

      2. add-sqr-sqrt 89.7%

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

      3. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 51.6%

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

      1. associate--l+ 95.1%

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

      2. +-inverses 95.1%

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

      3. metadata-eval 95.1%

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

   9. Simplified 52.0%

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

   if 5.00000000000000007e42 < z

   1. Initial program 83.8%

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

      1. associate-+l+ 83.8%

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

      2. associate-+l+ 83.8%

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

      3. +-commutative 83.8%

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

      4. +-commutative 83.8%

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

      5. associate-+l- 83.8%

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

      6. +-commutative 83.8%

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

      7. +-commutative 83.8%

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

   3. Simplified 83.8%

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

      1. flip-- 83.8%

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

      2. add-sqr-sqrt 67.0%

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

      3. +-commutative 67.0%

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

      4. add-sqr-sqrt 83.8%

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

      5. +-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. associate--l+ 88.9%

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

      2. +-inverses 88.9%

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

      3. metadata-eval 88.9%

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

   8. Simplified 88.9%

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

   9. Taylor expanded in z around inf 88.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + t\_1\right) + \left(\left(\left(y \cdot 0.5 + 1\right) - \sqrt{y}\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(t\_1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t))))
   (if (<= y 4.3e-6)
     (+
      (+ (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))) t_1)
      (+ (- (+ (* y 0.5) 1.0) (sqrt y)) 1.0))
     (+
      (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0))))
      (+ t_1 (- (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((t + 1.0)) - sqrt(t);
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_1) + ((((y * 0.5) + 1.0) - sqrt(y)) + 1.0);
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (t_1 + (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((t + 1.0d0)) - sqrt(t)
    if (y <= 4.3d-6) then
        tmp = ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + t_1) + ((((y * 0.5d0) + 1.0d0) - sqrt(y)) + 1.0d0)
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (t_1 + (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((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + t_1) + ((((y * 0.5) + 1.0) - Math.sqrt(y)) + 1.0);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (t_1 + (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((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if y <= 4.3e-6:
		tmp = ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + t_1) + ((((y * 0.5) + 1.0) - math.sqrt(y)) + 1.0)
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (t_1 + (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(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (y <= 4.3e-6)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + t_1) + Float64(Float64(Float64(Float64(y * 0.5) + 1.0) - sqrt(y)) + 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(t_1 + 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((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (y <= 4.3e-6)
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + t_1) + ((((y * 0.5) + 1.0) - sqrt(y)) + 1.0);
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (t_1 + (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[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.3e-6], N[(N[(N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(N[(N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + 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 y < 4.30000000000000033e-6

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. sub-neg 96.0%

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

      3. sub-neg 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 75.0%

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

      3. add-sqr-sqrt 97.0%

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

   7. Applied egg-rr 55.9%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in y around 0 56.1%

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

       1. *-commutative 56.1%

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

   12. Simplified 56.1%

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

   if 4.30000000000000033e-6 < y

   1. Initial program 79.3%

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

      1. associate-+l+ 79.3%

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

      2. associate-+l+ 79.3%

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

      3. +-commutative 79.3%

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

      4. +-commutative 79.3%

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

      5. associate-+l- 65.1%

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

      6. +-commutative 65.1%

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

      7. +-commutative 65.1%

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

   3. Simplified 65.1%

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

      1. flip-- 65.0%

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

      2. add-sqr-sqrt 54.4%

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

      3. +-commutative 54.4%

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

      4. add-sqr-sqrt 65.1%

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

      5. +-commutative 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. associate--l+ 69.9%

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

      2. +-inverses 69.9%

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

      3. metadata-eval 69.9%

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

   8. Simplified 69.9%

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

   9. Taylor expanded in z around inf 49.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\left(y \cdot 0.5 + 1\right) - \sqrt{y}\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\left(y \cdot 0.5 + 1\right) - \sqrt{y}\right) + 1\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;t\_1 + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 4.3e-6)
     (+
      (+ (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t)))
      (+ (- (+ (* y 0.5) 1.0) (sqrt y)) 1.0))
     (if (<= y 4.6e+24)
       (+ t_1 (- (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))) (sqrt x)))
       (/ 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((x + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((((y * 0.5) + 1.0) - sqrt(y)) + 1.0);
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) - sqrt(x));
	} else {
		tmp = 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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if (y <= 4.3d-6) then
        tmp = ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) + ((((y * 0.5d0) + 1.0d0) - sqrt(y)) + 1.0d0)
    else if (y <= 4.6d+24) then
        tmp = t_1 + ((1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y))) - sqrt(x))
    else
        tmp = 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((x + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) + ((((y * 0.5) + 1.0) - Math.sqrt(y)) + 1.0);
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y))) - Math.sqrt(x));
	} else {
		tmp = 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((x + 1.0))
	tmp = 0
	if y <= 4.3e-6:
		tmp = ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) + ((((y * 0.5) + 1.0) - math.sqrt(y)) + 1.0)
	elif y <= 4.6e+24:
		tmp = t_1 + ((1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y))) - math.sqrt(x))
	else:
		tmp = 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 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 4.3e-6)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) + Float64(Float64(Float64(Float64(y * 0.5) + 1.0) - sqrt(y)) + 1.0));
	elseif (y <= 4.6e+24)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y))) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(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((x + 1.0));
	tmp = 0.0;
	if (y <= 4.3e-6)
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) + ((((y * 0.5) + 1.0) - sqrt(y)) + 1.0);
	elseif (y <= 4.6e+24)
		tmp = t_1 + ((1.0 / (sqrt((y + 1.0)) + sqrt(y))) - sqrt(x));
	else
		tmp = 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.3e-6], N[(N[(N[(1.0 / 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] + N[(N[(N[(N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+24], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.30000000000000033e-6

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. sub-neg 96.0%

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

      3. sub-neg 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 75.0%

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

      3. add-sqr-sqrt 97.0%

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

   7. Applied egg-rr 55.9%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in y around 0 56.1%

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

       1. *-commutative 56.1%

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

   12. Simplified 56.1%

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

   if 4.30000000000000033e-6 < y < 4.5999999999999998e24

   1. Initial program 78.3%

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

      1. associate-+l+ 78.3%

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

      2. associate-+l+ 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-+l- 61.3%

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

      6. +-commutative 61.3%

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

      7. +-commutative 61.3%

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

   3. Simplified 61.3%

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

      1. flip-- 67.7%

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

      2. add-sqr-sqrt 58.7%

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

      3. add-sqr-sqrt 68.7%

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

   6. Applied egg-rr 68.4%

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

      1. associate--l+ 72.6%

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

      2. +-inverses 72.6%

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

      3. metadata-eval 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in z around inf 62.1%

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

   10. Taylor expanded in t around inf 21.2%

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

       1. associate--l+ 21.2%

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

   12. Simplified 21.2%

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

   if 4.5999999999999998e24 < y

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. associate-+r+ 79.4%

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

      3. associate-+r- 43.9%

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

      4. associate-+l- 18.1%

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

      5. associate-+r- 5.6%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 15.6%

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

      2. +-commutative 15.6%

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

      3. associate--l+ 19.2%

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

      4. +-commutative 19.2%

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

      5. associate-+r+ 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in z around inf 15.4%

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

      1. +-commutative 15.4%

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

   10. Simplified 15.4%

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

   11. Taylor expanded in y around inf 15.2%

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

       1. flip-- 15.2%

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

       2. add-sqr-sqrt 15.4%

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

       3. add-sqr-sqrt 15.2%

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

       4. +-commutative 15.2%

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

       5. +-commutative 15.2%

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

       6. +-commutative 15.2%

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

   13. Applied egg-rr 15.2%

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

       1. associate--l+ 21.4%

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

       2. +-inverses 21.4%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 21.4%

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

   15. Simplified 21.4%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) + \left(\left(\left(y \cdot 0.5 + 1\right) - \sqrt{y}\right) + 1\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 1.14 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 3.1 \cdot 10^{+224}\right):\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + \frac{1}{t\_1 + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= z 1.14e-32)
     (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
     (if (<= z 2.9e+17)
       (+ (- (+ t_1 (sqrt (+ z 1.0))) (+ (sqrt z) (sqrt y))) 1.0)
       (if (or (<= z 6.5e+196) (not (<= z 3.1e+224)))
         (+ (- t_2 (sqrt x)) (/ 1.0 (+ t_1 (sqrt y))))
         (/ 1.0 (+ (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((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (z <= 1.14e-32) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else if (z <= 2.9e+17) {
		tmp = ((t_1 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	} else if ((z <= 6.5e+196) || !(z <= 3.1e+224)) {
		tmp = (t_2 - sqrt(x)) + (1.0 / (t_1 + sqrt(y)));
	} else {
		tmp = 1.0 / (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((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (z <= 1.14d-32) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else if (z <= 2.9d+17) then
        tmp = ((t_1 + sqrt((z + 1.0d0))) - (sqrt(z) + sqrt(y))) + 1.0d0
    else if ((z <= 6.5d+196) .or. (.not. (z <= 3.1d+224))) then
        tmp = (t_2 - sqrt(x)) + (1.0d0 / (t_1 + sqrt(y)))
    else
        tmp = 1.0d0 / (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((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 1.14e-32) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 2.9e+17) {
		tmp = ((t_1 + Math.sqrt((z + 1.0))) - (Math.sqrt(z) + Math.sqrt(y))) + 1.0;
	} else if ((z <= 6.5e+196) || !(z <= 3.1e+224)) {
		tmp = (t_2 - Math.sqrt(x)) + (1.0 / (t_1 + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (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((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 1.14e-32:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	elif z <= 2.9e+17:
		tmp = ((t_1 + math.sqrt((z + 1.0))) - (math.sqrt(z) + math.sqrt(y))) + 1.0
	elif (z <= 6.5e+196) or not (z <= 3.1e+224):
		tmp = (t_2 - math.sqrt(x)) + (1.0 / (t_1 + math.sqrt(y)))
	else:
		tmp = 1.0 / (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(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 1.14e-32)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	elseif (z <= 2.9e+17)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(z + 1.0))) - Float64(sqrt(z) + sqrt(y))) + 1.0);
	elseif ((z <= 6.5e+196) || !(z <= 3.1e+224))
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(1.0 / Float64(t_1 + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(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((y + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 1.14e-32)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	elseif (z <= 2.9e+17)
		tmp = ((t_1 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	elseif ((z <= 6.5e+196) || ~((z <= 3.1e+224)))
		tmp = (t_2 - sqrt(x)) + (1.0 / (t_1 + sqrt(y)));
	else
		tmp = 1.0 / (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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.14e-32], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 2.9e+17], N[(N[(N[(t$95$1 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[Or[LessEqual[z, 6.5e+196], N[Not[LessEqual[z, 3.1e+224]], $MachinePrecision]], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.14000000000000005e-32

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 52.3%

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

   6. Taylor expanded in z around 0 21.9%

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

      1. associate--l+ 56.9%

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

      2. +-commutative 56.9%

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

      3. associate--l+ 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in y around 0 24.4%

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

       1. associate--l+ 36.0%

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

   11. Simplified 36.0%

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

   if 1.14000000000000005e-32 < z < 2.9e17

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-+r+ 89.0%

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

      3. associate-+r- 74.8%

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

      4. associate-+l- 64.1%

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

      5. associate-+r- 60.1%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in t around inf 31.4%

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

      1. associate--l+ 34.6%

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

      2. +-commutative 34.6%

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

      3. associate--l+ 36.8%

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

      4. +-commutative 36.8%

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

      5. associate-+r+ 36.8%

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

   7. Simplified 36.8%

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

   8. Taylor expanded in x around 0 44.4%

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

      1. associate--l+ 49.7%

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

      2. +-commutative 49.7%

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

   10. Simplified 49.7%

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

   if 2.9e17 < z < 6.49999999999999968e196 or 3.0999999999999999e224 < z

   1. Initial program 80.8%

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

      1. associate-+l+ 80.8%

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

      2. associate-+l+ 80.8%

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

      3. +-commutative 80.8%

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

      4. +-commutative 80.8%

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

      5. associate-+l- 80.8%

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

      6. +-commutative 80.8%

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

      7. +-commutative 80.8%

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

   3. Simplified 80.8%

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

      1. flip-- 86.6%

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

      2. add-sqr-sqrt 61.4%

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

      3. add-sqr-sqrt 86.7%

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

   6. Applied egg-rr 81.6%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in z around inf 84.3%

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

   10. Taylor expanded in t around inf 39.6%

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

   if 6.49999999999999968e196 < z < 3.0999999999999999e224

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-+r+ 89.3%

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

      3. associate-+r- 80.2%

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

      4. associate-+l- 78.5%

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

      5. associate-+r- 78.5%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 10.6%

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

      2. +-commutative 10.6%

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

      3. associate--l+ 10.6%

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

      4. +-commutative 10.6%

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

      5. associate-+r+ 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in z around inf 28.3%

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

      1. +-commutative 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in y around inf 10.0%

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

       1. flip-- 10.0%

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

       2. add-sqr-sqrt 9.6%

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

       3. add-sqr-sqrt 10.0%

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

       4. +-commutative 10.0%

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

       5. +-commutative 10.0%

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

       6. +-commutative 10.0%

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

   13. Applied egg-rr 10.0%

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

       1. associate--l+ 11.5%

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

       2. +-inverses 11.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 11.5%

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

   15. Simplified 11.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.14 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 3.1 \cdot 10^{+224}\right):\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{z + 1}\right) + 1\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 3.1 \cdot 10^{+224}\right):\\ \;\;\;\;\left(t\_2 - \sqrt{x}\right) + \frac{1}{t\_1 + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= z 1.55e-30)
     (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
     (if (<= z 2.7e+17)
       (- (+ (+ t_1 (sqrt (+ z 1.0))) 1.0) (+ (sqrt z) (sqrt y)))
       (if (or (<= z 6.5e+196) (not (<= z 3.1e+224)))
         (+ (- t_2 (sqrt x)) (/ 1.0 (+ t_1 (sqrt y))))
         (/ 1.0 (+ (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((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (z <= 1.55e-30) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else if (z <= 2.7e+17) {
		tmp = ((t_1 + sqrt((z + 1.0))) + 1.0) - (sqrt(z) + sqrt(y));
	} else if ((z <= 6.5e+196) || !(z <= 3.1e+224)) {
		tmp = (t_2 - sqrt(x)) + (1.0 / (t_1 + sqrt(y)));
	} else {
		tmp = 1.0 / (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((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if (z <= 1.55d-30) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else if (z <= 2.7d+17) then
        tmp = ((t_1 + sqrt((z + 1.0d0))) + 1.0d0) - (sqrt(z) + sqrt(y))
    else if ((z <= 6.5d+196) .or. (.not. (z <= 3.1d+224))) then
        tmp = (t_2 - sqrt(x)) + (1.0d0 / (t_1 + sqrt(y)))
    else
        tmp = 1.0d0 / (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((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 1.55e-30) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 2.7e+17) {
		tmp = ((t_1 + Math.sqrt((z + 1.0))) + 1.0) - (Math.sqrt(z) + Math.sqrt(y));
	} else if ((z <= 6.5e+196) || !(z <= 3.1e+224)) {
		tmp = (t_2 - Math.sqrt(x)) + (1.0 / (t_1 + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (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((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 1.55e-30:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	elif z <= 2.7e+17:
		tmp = ((t_1 + math.sqrt((z + 1.0))) + 1.0) - (math.sqrt(z) + math.sqrt(y))
	elif (z <= 6.5e+196) or not (z <= 3.1e+224):
		tmp = (t_2 - math.sqrt(x)) + (1.0 / (t_1 + math.sqrt(y)))
	else:
		tmp = 1.0 / (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(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 1.55e-30)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	elseif (z <= 2.7e+17)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(z + 1.0))) + 1.0) - Float64(sqrt(z) + sqrt(y)));
	elseif ((z <= 6.5e+196) || !(z <= 3.1e+224))
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(1.0 / Float64(t_1 + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(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((y + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 1.55e-30)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	elseif (z <= 2.7e+17)
		tmp = ((t_1 + sqrt((z + 1.0))) + 1.0) - (sqrt(z) + sqrt(y));
	elseif ((z <= 6.5e+196) || ~((z <= 3.1e+224)))
		tmp = (t_2 - sqrt(x)) + (1.0 / (t_1 + sqrt(y)));
	else
		tmp = 1.0 / (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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.55e-30], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 2.7e+17], N[(N[(N[(t$95$1 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 6.5e+196], N[Not[LessEqual[z, 3.1e+224]], $MachinePrecision]], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.54999999999999995e-30

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 52.8%

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

   6. Taylor expanded in z around 0 21.7%

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

      1. associate--l+ 56.6%

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

      2. +-commutative 56.6%

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

      3. associate--l+ 49.1%

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

   8. Simplified 49.1%

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

   9. Taylor expanded in y around 0 24.4%

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

       1. associate--l+ 35.9%

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

   11. Simplified 35.9%

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

   if 1.54999999999999995e-30 < z < 2.7e17

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-+r+ 88.6%

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

      3. associate-+r- 77.1%

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

      4. associate-+l- 66.0%

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

      5. associate-+r- 62.5%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in t around inf 32.6%

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

      1. associate--l+ 35.3%

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

      2. +-commutative 35.3%

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

      3. associate--l+ 37.5%

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

      4. +-commutative 37.5%

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

      5. associate-+r+ 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in x around 0 46.1%

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

   if 2.7e17 < z < 6.49999999999999968e196 or 3.0999999999999999e224 < z

   1. Initial program 80.8%

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

      1. associate-+l+ 80.8%

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

      2. associate-+l+ 80.8%

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

      3. +-commutative 80.8%

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

      4. +-commutative 80.8%

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

      5. associate-+l- 80.8%

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

      6. +-commutative 80.8%

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

      7. +-commutative 80.8%

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

   3. Simplified 80.8%

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

      1. flip-- 86.6%

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

      2. add-sqr-sqrt 61.4%

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

      3. add-sqr-sqrt 86.7%

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

   6. Applied egg-rr 81.6%

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

      1. associate--l+ 89.3%

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

      2. +-inverses 89.3%

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

      3. metadata-eval 89.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in z around inf 84.3%

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

   10. Taylor expanded in t around inf 39.6%

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

   if 6.49999999999999968e196 < z < 3.0999999999999999e224

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-+r+ 89.3%

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

      3. associate-+r- 80.2%

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

      4. associate-+l- 78.5%

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

      5. associate-+r- 78.5%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 10.6%

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

      2. +-commutative 10.6%

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

      3. associate--l+ 10.6%

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

      4. +-commutative 10.6%

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

      5. associate-+r+ 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in z around inf 28.3%

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

      1. +-commutative 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in y around inf 10.0%

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

       1. flip-- 10.0%

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

       2. add-sqr-sqrt 9.6%

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

       3. add-sqr-sqrt 10.0%

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

       4. +-commutative 10.0%

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

       5. +-commutative 10.0%

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

       6. +-commutative 10.0%

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

   13. Applied egg-rr 10.0%

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

       1. associate--l+ 11.5%

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

       2. +-inverses 11.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 11.5%

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

   15. Simplified 11.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{z + 1}\right) + 1\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 3.1 \cdot 10^{+224}\right):\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 1.18 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(t\_2 + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196}:\\ \;\;\;\;t\_1 + \left(\frac{1}{t\_2 + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 - \sqrt{y}\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= z 1.18e-32)
     (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
     (if (<= z 5e+17)
       (+ (- (+ t_2 (sqrt (+ z 1.0))) (+ (sqrt z) (sqrt y))) 1.0)
       (if (<= z 6.5e+196)
         (+ t_1 (- (/ 1.0 (+ t_2 (sqrt y))) (sqrt x)))
         (if (<= z 4.6e+234)
           (/ 1.0 (+ (sqrt x) t_1))
           (+ (- t_2 (sqrt y)) 1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (z <= 1.18e-32) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else if (z <= 5e+17) {
		tmp = ((t_2 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	} else if (z <= 6.5e+196) {
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	} else if (z <= 4.6e+234) {
		tmp = 1.0 / (sqrt(x) + t_1);
	} else {
		tmp = (t_2 - sqrt(y)) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (z <= 1.18d-32) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else if (z <= 5d+17) then
        tmp = ((t_2 + sqrt((z + 1.0d0))) - (sqrt(z) + sqrt(y))) + 1.0d0
    else if (z <= 6.5d+196) then
        tmp = t_1 + ((1.0d0 / (t_2 + sqrt(y))) - sqrt(x))
    else if (z <= 4.6d+234) then
        tmp = 1.0d0 / (sqrt(x) + t_1)
    else
        tmp = (t_2 - sqrt(y)) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 1.18e-32) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 5e+17) {
		tmp = ((t_2 + Math.sqrt((z + 1.0))) - (Math.sqrt(z) + Math.sqrt(y))) + 1.0;
	} else if (z <= 6.5e+196) {
		tmp = t_1 + ((1.0 / (t_2 + Math.sqrt(y))) - Math.sqrt(x));
	} else if (z <= 4.6e+234) {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	} else {
		tmp = (t_2 - Math.sqrt(y)) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 1.18e-32:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	elif z <= 5e+17:
		tmp = ((t_2 + math.sqrt((z + 1.0))) - (math.sqrt(z) + math.sqrt(y))) + 1.0
	elif z <= 6.5e+196:
		tmp = t_1 + ((1.0 / (t_2 + math.sqrt(y))) - math.sqrt(x))
	elif z <= 4.6e+234:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	else:
		tmp = (t_2 - math.sqrt(y)) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 1.18e-32)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	elseif (z <= 5e+17)
		tmp = Float64(Float64(Float64(t_2 + sqrt(Float64(z + 1.0))) - Float64(sqrt(z) + sqrt(y))) + 1.0);
	elseif (z <= 6.5e+196)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) - sqrt(x)));
	elseif (z <= 4.6e+234)
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	else
		tmp = Float64(Float64(t_2 - sqrt(y)) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 1.18e-32)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	elseif (z <= 5e+17)
		tmp = ((t_2 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	elseif (z <= 6.5e+196)
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	elseif (z <= 4.6e+234)
		tmp = 1.0 / (sqrt(x) + t_1);
	else
		tmp = (t_2 - sqrt(y)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.18e-32], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 5e+17], N[(N[(N[(t$95$2 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.5e+196], N[(t$95$1 + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e+234], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < 1.17999999999999997e-32

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 52.3%

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

   6. Taylor expanded in z around 0 21.9%

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

      1. associate--l+ 56.9%

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

      2. +-commutative 56.9%

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

      3. associate--l+ 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in y around 0 24.4%

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

       1. associate--l+ 36.0%

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

   11. Simplified 36.0%

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

   if 1.17999999999999997e-32 < z < 5e17

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-+r+ 89.0%

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

      3. associate-+r- 74.8%

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

      4. associate-+l- 64.1%

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

      5. associate-+r- 60.1%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in t around inf 31.4%

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

      1. associate--l+ 34.6%

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

      2. +-commutative 34.6%

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

      3. associate--l+ 36.8%

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

      4. +-commutative 36.8%

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

      5. associate-+r+ 36.8%

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

   7. Simplified 36.8%

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

   8. Taylor expanded in x around 0 44.4%

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

      1. associate--l+ 49.7%

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

      2. +-commutative 49.7%

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

   10. Simplified 49.7%

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

   if 5e17 < z < 6.49999999999999968e196

   1. Initial program 78.0%

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

      1. associate-+l+ 78.0%

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

      2. associate-+l+ 78.0%

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

      3. +-commutative 78.0%

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

      4. +-commutative 78.0%

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

      5. associate-+l- 78.0%

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

      6. +-commutative 78.0%

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

      7. +-commutative 78.0%

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

   3. Simplified 78.0%

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

      1. flip-- 83.0%

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

      2. add-sqr-sqrt 61.6%

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

      3. add-sqr-sqrt 83.2%

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

   6. Applied egg-rr 79.0%

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

      1. associate--l+ 85.9%

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

      2. +-inverses 85.9%

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

      3. metadata-eval 85.9%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in z around inf 81.5%

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

   10. Taylor expanded in t around inf 26.1%

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

       1. associate--l+ 26.1%

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

   12. Simplified 26.1%

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

   if 6.49999999999999968e196 < z < 4.6000000000000002e234

   1. Initial program 93.1%

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

      1. +-commutative 93.1%

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

      2. associate-+r+ 93.1%

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

      3. associate-+r- 87.3%

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

      4. associate-+l- 72.3%

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

      5. associate-+r- 72.3%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 11.2%

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

      2. +-commutative 11.2%

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

      3. associate--l+ 11.2%

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

      4. +-commutative 11.2%

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

      5. associate-+r+ 11.2%

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

   7. Simplified 11.2%

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

   8. Taylor expanded in z around inf 22.2%

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

      1. +-commutative 22.2%

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

   10. Simplified 22.2%

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

   11. Taylor expanded in y around inf 9.6%

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

       1. flip-- 9.6%

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

       2. add-sqr-sqrt 10.3%

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

       3. add-sqr-sqrt 9.6%

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

       4. +-commutative 9.6%

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

       5. +-commutative 9.6%

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

       6. +-commutative 9.6%

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

   13. Applied egg-rr 9.6%

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

       1. associate--l+ 11.8%

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

       2. +-inverses 11.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 11.8%

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

   15. Simplified 11.8%

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

   if 4.6000000000000002e234 < z

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

      2. associate-+r+ 84.1%

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

      3. associate-+r- 68.3%

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

      4. associate-+l- 49.4%

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

      5. associate-+r- 49.4%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 18.3%

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

      2. +-commutative 18.3%

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

      3. associate--l+ 18.3%

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

      4. +-commutative 18.3%

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

      5. associate-+r+ 18.3%

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

   7. Simplified 18.3%

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

   8. Taylor expanded in z around inf 26.5%

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

      1. +-commutative 26.5%

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

   10. Simplified 26.5%

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

   11. Taylor expanded in x around 0 23.2%

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

       1. associate--l+ 50.4%

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

   13. Simplified 50.4%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 1.14 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 4.6 \cdot 10^{+234}\right):\\ \;\;\;\;\left(t\_1 - \sqrt{y}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 1.14e-32)
     (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
     (if (<= z 2.9e+17)
       (+ (- (+ t_1 (sqrt (+ z 1.0))) (+ (sqrt z) (sqrt y))) 1.0)
       (if (or (<= z 6.5e+196) (not (<= z 4.6e+234)))
         (+ (- t_1 (sqrt y)) 1.0)
         (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 1.14e-32) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else if (z <= 2.9e+17) {
		tmp = ((t_1 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	} else if ((z <= 6.5e+196) || !(z <= 4.6e+234)) {
		tmp = (t_1 - sqrt(y)) + 1.0;
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 1.14d-32) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else if (z <= 2.9d+17) then
        tmp = ((t_1 + sqrt((z + 1.0d0))) - (sqrt(z) + sqrt(y))) + 1.0d0
    else if ((z <= 6.5d+196) .or. (.not. (z <= 4.6d+234))) then
        tmp = (t_1 - sqrt(y)) + 1.0d0
    else
        tmp = 1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 1.14e-32) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 2.9e+17) {
		tmp = ((t_1 + Math.sqrt((z + 1.0))) - (Math.sqrt(z) + Math.sqrt(y))) + 1.0;
	} else if ((z <= 6.5e+196) || !(z <= 4.6e+234)) {
		tmp = (t_1 - Math.sqrt(y)) + 1.0;
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 1.14e-32:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	elif z <= 2.9e+17:
		tmp = ((t_1 + math.sqrt((z + 1.0))) - (math.sqrt(z) + math.sqrt(y))) + 1.0
	elif (z <= 6.5e+196) or not (z <= 4.6e+234):
		tmp = (t_1 - math.sqrt(y)) + 1.0
	else:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 1.14e-32)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	elseif (z <= 2.9e+17)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(z + 1.0))) - Float64(sqrt(z) + sqrt(y))) + 1.0);
	elseif ((z <= 6.5e+196) || !(z <= 4.6e+234))
		tmp = Float64(Float64(t_1 - sqrt(y)) + 1.0);
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 1.14e-32)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	elseif (z <= 2.9e+17)
		tmp = ((t_1 + sqrt((z + 1.0))) - (sqrt(z) + sqrt(y))) + 1.0;
	elseif ((z <= 6.5e+196) || ~((z <= 4.6e+234)))
		tmp = (t_1 - sqrt(y)) + 1.0;
	else
		tmp = 1.0 / (sqrt(x) + sqrt((x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.14e-32], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 2.9e+17], N[(N[(N[(t$95$1 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[Or[LessEqual[z, 6.5e+196], N[Not[LessEqual[z, 4.6e+234]], $MachinePrecision]], N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.14000000000000005e-32

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

      6. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 52.3%

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

   6. Taylor expanded in z around 0 21.9%

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

      1. associate--l+ 56.9%

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

      2. +-commutative 56.9%

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

      3. associate--l+ 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in y around 0 24.4%

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

       1. associate--l+ 36.0%

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

   11. Simplified 36.0%

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

   if 1.14000000000000005e-32 < z < 2.9e17

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-+r+ 89.0%

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

      3. associate-+r- 74.8%

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

      4. associate-+l- 64.1%

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

      5. associate-+r- 60.1%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in t around inf 31.4%

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

      1. associate--l+ 34.6%

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

      2. +-commutative 34.6%

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

      3. associate--l+ 36.8%

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

      4. +-commutative 36.8%

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

      5. associate-+r+ 36.8%

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

   7. Simplified 36.8%

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

   8. Taylor expanded in x around 0 44.4%

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

      1. associate--l+ 49.7%

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

      2. +-commutative 49.7%

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

   10. Simplified 49.7%

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

   if 2.9e17 < z < 6.49999999999999968e196 or 4.6000000000000002e234 < z

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-+r+ 80.0%

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

      3. associate-+r- 63.6%

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

      4. associate-+l- 52.0%

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

      5. associate-+r- 52.0%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in t around inf 4.0%

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

      1. associate--l+ 16.4%

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

      2. +-commutative 16.4%

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

      3. associate--l+ 13.2%

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

      4. +-commutative 13.2%

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

      5. associate-+r+ 13.1%

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

   7. Simplified 13.1%

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

   8. Taylor expanded in z around inf 25.6%

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

      1. +-commutative 25.6%

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

   10. Simplified 25.6%

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

   11. Taylor expanded in x around 0 31.0%

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

       1. associate--l+ 48.4%

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

   13. Simplified 48.4%

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

   if 6.49999999999999968e196 < z < 4.6000000000000002e234

   1. Initial program 93.1%

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

      1. +-commutative 93.1%

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

      2. associate-+r+ 93.1%

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

      3. associate-+r- 87.3%

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

      4. associate-+l- 72.3%

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

      5. associate-+r- 72.3%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around inf 3.1%

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

      1. associate--l+ 11.2%

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

      2. +-commutative 11.2%

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

      3. associate--l+ 11.2%

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

      4. +-commutative 11.2%

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

      5. associate-+r+ 11.2%

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

   7. Simplified 11.2%

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

   8. Taylor expanded in z around inf 22.2%

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

      1. +-commutative 22.2%

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

   10. Simplified 22.2%

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

   11. Taylor expanded in y around inf 9.6%

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

       1. flip-- 9.6%

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

       2. add-sqr-sqrt 10.3%

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

       3. add-sqr-sqrt 9.6%

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

       4. +-commutative 9.6%

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

       5. +-commutative 9.6%

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

       6. +-commutative 9.6%

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

   13. Applied egg-rr 9.6%

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

       1. associate--l+ 11.8%

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

       2. +-inverses 11.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 11.8%

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

   15. Simplified 11.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.14 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{z + 1}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + 1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 4.6 \cdot 10^{+234}\right):\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(t\_2 + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) - \sqrt{y}\right) + 1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;t\_1 + \left(\frac{1}{t\_2 + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 4.3e-6)
     (+ (- (+ t_2 (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (sqrt y)) 1.0)
     (if (<= y 4.6e+24)
       (+ t_1 (- (/ 1.0 (+ t_2 (sqrt y))) (sqrt x)))
       (/ 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((x + 1.0));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((t_2 + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) - sqrt(y)) + 1.0;
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	} else {
		tmp = 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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (y <= 4.3d-6) then
        tmp = ((t_2 + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) - sqrt(y)) + 1.0d0
    else if (y <= 4.6d+24) then
        tmp = t_1 + ((1.0d0 / (t_2 + sqrt(y))) - sqrt(x))
    else
        tmp = 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((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((t_2 + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) - Math.sqrt(y)) + 1.0;
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (t_2 + Math.sqrt(y))) - Math.sqrt(x));
	} else {
		tmp = 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((x + 1.0))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 4.3e-6:
		tmp = ((t_2 + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) - math.sqrt(y)) + 1.0
	elif y <= 4.6e+24:
		tmp = t_1 + ((1.0 / (t_2 + math.sqrt(y))) - math.sqrt(x))
	else:
		tmp = 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 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 4.3e-6)
		tmp = Float64(Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) - sqrt(y)) + 1.0);
	elseif (y <= 4.6e+24)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(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((x + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 4.3e-6)
		tmp = ((t_2 + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) - sqrt(y)) + 1.0;
	elseif (y <= 4.6e+24)
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	else
		tmp = 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.3e-6], N[(N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 4.6e+24], N[(t$95$1 + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.30000000000000033e-6

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. sub-neg 96.0%

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

      3. sub-neg 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 75.0%

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

      3. add-sqr-sqrt 97.0%

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

   7. Applied egg-rr 55.9%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in t around inf 55.6%

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

       1. associate--l+ 55.6%

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

       2. +-commutative 55.6%

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

   12. Simplified 55.6%

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

   if 4.30000000000000033e-6 < y < 4.5999999999999998e24

   1. Initial program 78.3%

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

      1. associate-+l+ 78.3%

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

      2. associate-+l+ 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-+l- 61.3%

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

      6. +-commutative 61.3%

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

      7. +-commutative 61.3%

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

   3. Simplified 61.3%

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

      1. flip-- 67.7%

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

      2. add-sqr-sqrt 58.7%

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

      3. add-sqr-sqrt 68.7%

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

   6. Applied egg-rr 68.4%

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

      1. associate--l+ 72.6%

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

      2. +-inverses 72.6%

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

      3. metadata-eval 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in z around inf 62.1%

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

   10. Taylor expanded in t around inf 21.2%

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

       1. associate--l+ 21.2%

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

   12. Simplified 21.2%

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

   if 4.5999999999999998e24 < y

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. associate-+r+ 79.4%

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

      3. associate-+r- 43.9%

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

      4. associate-+l- 18.1%

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

      5. associate-+r- 5.6%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 15.6%

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

      2. +-commutative 15.6%

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

      3. associate--l+ 19.2%

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

      4. +-commutative 19.2%

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

      5. associate-+r+ 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in z around inf 15.4%

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

      1. +-commutative 15.4%

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

   10. Simplified 15.4%

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

   11. Taylor expanded in y around inf 15.2%

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

       1. flip-- 15.2%

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

       2. add-sqr-sqrt 15.4%

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

       3. add-sqr-sqrt 15.2%

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

       4. +-commutative 15.2%

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

       5. +-commutative 15.2%

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

       6. +-commutative 15.2%

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

   13. Applied egg-rr 15.2%

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

       1. associate--l+ 21.4%

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

       2. +-inverses 21.4%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 21.4%

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

   15. Simplified 21.4%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) - \sqrt{y}\right) + 1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(t\_2 + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + 1\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;t\_1 + \left(\frac{1}{t\_2 + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))) (t_2 (sqrt (+ y 1.0))))
   (if (<= y 4.3e-6)
     (- (+ (+ t_2 (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) 1.0) (sqrt y))
     (if (<= y 4.6e+24)
       (+ t_1 (- (/ 1.0 (+ t_2 (sqrt y))) (sqrt x)))
       (/ 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((x + 1.0));
	double t_2 = sqrt((y + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((t_2 + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + 1.0) - sqrt(y);
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	} else {
		tmp = 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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    if (y <= 4.3d-6) then
        tmp = ((t_2 + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + 1.0d0) - sqrt(y)
    else if (y <= 4.6d+24) then
        tmp = t_1 + ((1.0d0 / (t_2 + sqrt(y))) - sqrt(x))
    else
        tmp = 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((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 4.3e-6) {
		tmp = ((t_2 + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + 1.0) - Math.sqrt(y);
	} else if (y <= 4.6e+24) {
		tmp = t_1 + ((1.0 / (t_2 + Math.sqrt(y))) - Math.sqrt(x));
	} else {
		tmp = 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((x + 1.0))
	t_2 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 4.3e-6:
		tmp = ((t_2 + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + 1.0) - math.sqrt(y)
	elif y <= 4.6e+24:
		tmp = t_1 + ((1.0 / (t_2 + math.sqrt(y))) - math.sqrt(x))
	else:
		tmp = 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 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 4.3e-6)
		tmp = Float64(Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + 1.0) - sqrt(y));
	elseif (y <= 4.6e+24)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(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((x + 1.0));
	t_2 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 4.3e-6)
		tmp = ((t_2 + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + 1.0) - sqrt(y);
	elseif (y <= 4.6e+24)
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) - sqrt(x));
	else
		tmp = 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.3e-6], N[(N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+24], N[(t$95$1 + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.30000000000000033e-6

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. sub-neg 96.0%

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

      3. sub-neg 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. flip-- 96.6%

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

      2. add-sqr-sqrt 75.0%

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

      3. add-sqr-sqrt 97.0%

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

   7. Applied egg-rr 55.9%

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

      1. associate--l+ 97.6%

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

      2. +-inverses 97.6%

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

      3. metadata-eval 97.6%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in t around inf 55.6%

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

   if 4.30000000000000033e-6 < y < 4.5999999999999998e24

   1. Initial program 78.3%

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

      1. associate-+l+ 78.3%

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

      2. associate-+l+ 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-+l- 61.3%

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

      6. +-commutative 61.3%

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

      7. +-commutative 61.3%

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

   3. Simplified 61.3%

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

      1. flip-- 67.7%

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

      2. add-sqr-sqrt 58.7%

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

      3. add-sqr-sqrt 68.7%

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

   6. Applied egg-rr 68.4%

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

      1. associate--l+ 72.6%

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

      2. +-inverses 72.6%

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

      3. metadata-eval 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in z around inf 62.1%

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

   10. Taylor expanded in t around inf 21.2%

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

       1. associate--l+ 21.2%

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

   12. Simplified 21.2%

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

   if 4.5999999999999998e24 < y

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. associate-+r+ 79.4%

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

      3. associate-+r- 43.9%

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

      4. associate-+l- 18.1%

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

      5. associate-+r- 5.6%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 15.6%

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

      2. +-commutative 15.6%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 19.2%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 19.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

      5. associate-+r+ 19.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

   7. Simplified 19.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 15.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 15.4%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   10. Simplified 15.4%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   11. Taylor expanded in y around inf 15.2%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   12. Step-by-step derivation

       1. flip-- 15.2%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 15.4%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 15.2%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 15.2%

          \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} \]

       5. +-commutative 15.2%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} \]

       6. +-commutative 15.2%

          \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} \]

   13. Applied egg-rr 15.2%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   14. Step-by-step derivation

       1. associate--l+ 21.4%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 21.4%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 21.4%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   15. Simplified 21.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + 1\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.34:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 4.6 \cdot 10^{+234}\right):\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.34)
   (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
   (if (or (<= z 6.5e+196) (not (<= z 4.6e+234)))
     (+ (- (sqrt (+ y 1.0)) (sqrt y)) 1.0)
     (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.34) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else if ((z <= 6.5e+196) || !(z <= 4.6e+234)) {
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.34d0) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else if ((z <= 6.5d+196) .or. (.not. (z <= 4.6d+234))) then
        tmp = (sqrt((y + 1.0d0)) - sqrt(y)) + 1.0d0
    else
        tmp = 1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.34) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else if ((z <= 6.5e+196) || !(z <= 4.6e+234)) {
		tmp = (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + 1.0;
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.34:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	elif (z <= 6.5e+196) or not (z <= 4.6e+234):
		tmp = (math.sqrt((y + 1.0)) - math.sqrt(y)) + 1.0
	else:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.34)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	elseif ((z <= 6.5e+196) || !(z <= 4.6e+234))
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + 1.0);
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.34)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	elseif ((z <= 6.5e+196) || ~((z <= 4.6e+234)))
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
	else
		tmp = 1.0 / (sqrt(x) + sqrt((x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.34], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[Or[LessEqual[z, 6.5e+196], N[Not[LessEqual[z, 4.6e+234]], $MachinePrecision]], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 0.340000000000000024

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \left(\color{blue}{1} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 19.8%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 53.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + y}\right) - \left(\sqrt{t} + \sqrt{y}\right)\right)} \]

      2. +-commutative 53.9%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

      3. associate--l+ 46.6%

         \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   9. Taylor expanded in y around 0 22.6%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \sqrt{t}} \]
   10. Step-by-step derivation

       1. associate--l+ 34.4%

          \[\leadsto \color{blue}{3 + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]

   11. Simplified 34.4%

       \[\leadsto \color{blue}{3 + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]

   if 0.340000000000000024 < z < 6.49999999999999968e196 or 4.6000000000000002e234 < z

   1. Initial program 79.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 79.8%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 63.5%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 51.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 51.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   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+ 16.4%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 16.4%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 13.3%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 13.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

      5. associate-+r+ 13.3%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

   7. Simplified 13.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 24.8%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 24.8%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   10. Simplified 24.8%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   11. Taylor expanded in x around 0 30.7%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   12. Step-by-step derivation

       1. associate--l+ 47.5%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   13. Simplified 47.5%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 6.49999999999999968e196 < z < 4.6000000000000002e234

   1. Initial program 93.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 93.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 93.1%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 87.3%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 72.3%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 72.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 11.2%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 11.2%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 11.2%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 11.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

      5. associate-+r+ 11.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

   7. Simplified 11.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 22.2%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 22.2%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   10. Simplified 22.2%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   11. Taylor expanded in y around inf 9.6%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   12. Step-by-step derivation

       1. flip-- 9.6%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 10.3%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. add-sqr-sqrt 9.6%

          \[\leadsto \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 9.6%

          \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} \]

       5. +-commutative 9.6%

          \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} \]

       6. +-commutative 9.6%

          \[\leadsto \frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} \]

   13. Applied egg-rr 9.6%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}} \]
   14. Step-by-step derivation

       1. associate--l+ 11.8%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

       2. +-inverses 11.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x} + \sqrt{1 + x}} \]

       3. metadata-eval 11.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{x} + \sqrt{1 + x}} \]

   15. Simplified 11.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.34:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+196} \lor \neg \left(z \leq 4.6 \cdot 10^{+234}\right):\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;z \leq 0.62:\\ \;\;\;\;t\_1 + 2\\ \mathbf{else}:\\ \;\;\;\;t\_1 + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= z 0.62) (+ t_1 2.0) (+ t_1 1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (z <= 0.62) {
		tmp = t_1 + 2.0;
	} else {
		tmp = t_1 + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0)) - sqrt(y)
    if (z <= 0.62d0) then
        tmp = t_1 + 2.0d0
    else
        tmp = t_1 + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (z <= 0.62) {
		tmp = t_1 + 2.0;
	} else {
		tmp = t_1 + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if z <= 0.62:
		tmp = t_1 + 2.0
	else:
		tmp = t_1 + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (z <= 0.62)
		tmp = Float64(t_1 + 2.0);
	else
		tmp = Float64(t_1 + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (z <= 0.62)
		tmp = t_1 + 2.0;
	else
		tmp = t_1 + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 0.62], N[(t$95$1 + 2.0), $MachinePrecision], N[(t$95$1 + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 0.619999999999999996

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \left(\color{blue}{1} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 19.8%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 53.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + y}\right) - \left(\sqrt{t} + \sqrt{y}\right)\right)} \]

      2. +-commutative 53.9%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

      3. associate--l+ 46.6%

         \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   9. Taylor expanded in t around inf 55.0%

      \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 0.619999999999999996 < z

   1. Initial program 81.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 81.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 81.1%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 65.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 53.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 32.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 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+ 15.9%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 15.9%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 13.1%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 13.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

      5. associate-+r+ 13.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

   7. Simplified 13.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 24.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 24.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   10. Simplified 24.6%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   11. Taylor expanded in x around 0 31.4%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   12. Step-by-step derivation

       1. associate--l+ 47.8%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   13. Simplified 47.8%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.62:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.31:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.31)
   (+ (- (sqrt (+ t 1.0)) (sqrt t)) 3.0)
   (+ (- (sqrt (+ y 1.0)) (sqrt y)) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.31) {
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	} else {
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.31d0) then
        tmp = (sqrt((t + 1.0d0)) - sqrt(t)) + 3.0d0
    else
        tmp = (sqrt((y + 1.0d0)) - sqrt(y)) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.31) {
		tmp = (Math.sqrt((t + 1.0)) - Math.sqrt(t)) + 3.0;
	} else {
		tmp = (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.31:
		tmp = (math.sqrt((t + 1.0)) - math.sqrt(t)) + 3.0
	else:
		tmp = (math.sqrt((y + 1.0)) - math.sqrt(y)) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.31)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) - sqrt(t)) + 3.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.31)
		tmp = (sqrt((t + 1.0)) - sqrt(t)) + 3.0;
	else
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.31], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.309999999999999998

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 96.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \left(\color{blue}{1} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in z around 0 19.8%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 53.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + y}\right) - \left(\sqrt{t} + \sqrt{y}\right)\right)} \]

      2. +-commutative 53.9%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{y}\right)\right) \]

      3. associate--l+ 46.6%

         \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + y} + \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{y}\right)\right)\right)} \]

   9. Taylor expanded in y around 0 22.6%

      \[\leadsto \color{blue}{\left(3 + \sqrt{1 + t}\right) - \sqrt{t}} \]
   10. Step-by-step derivation

       1. associate--l+ 34.4%

          \[\leadsto \color{blue}{3 + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]

   11. Simplified 34.4%

       \[\leadsto \color{blue}{3 + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]

   if 0.309999999999999998 < z

   1. Initial program 81.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 81.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ 81.1%

         \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      3. associate-+r- 65.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 53.8%

         \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. associate-+r- 53.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. Simplified 32.5%

      \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 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+ 15.9%

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative 15.9%

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. associate--l+ 13.1%

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative 13.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

      5. associate-+r+ 13.1%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

   7. Simplified 13.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf 24.6%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 24.6%

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   10. Simplified 24.6%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   11. Taylor expanded in x around 0 31.4%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   12. Step-by-step derivation

       1. associate--l+ 47.8%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   13. Simplified 47.8%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.31:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{y + 1} - \sqrt{y}\right) + 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (- (sqrt (+ y 1.0)) (sqrt y)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((y + 1.0d0)) - sqrt(y)) + 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((y + 1.0)) - math.sqrt(y)) + 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + 1.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((y + 1.0)) - sqrt(y)) + 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 87.8%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 72.4%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 61.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.0%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 18.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 18.9%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. associate--l+ 20.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   5. associate-+r+ 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

7. Simplified 20.4%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in z around inf 18.7%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 18.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

10. Simplified 18.7%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

11. Taylor expanded in x around 0 27.6%

    \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
12. Step-by-step derivation

    1. associate--l+ 42.0%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

13. Simplified 42.0%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

14. Final simplification 42.0%

    \[\leadsto \left(\sqrt{y + 1} - \sqrt{y}\right) + 1 \]
15. Add Preprocessing

Alternative 17: 36.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0)) - sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 87.8%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 72.4%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 61.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.0%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 18.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 18.9%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. associate--l+ 20.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   5. associate-+r+ 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

7. Simplified 20.4%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in z around inf 18.7%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 18.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

10. Simplified 18.7%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

11. Taylor expanded in y around inf 13.0%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

12. Final simplification 13.0%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]
13. Add Preprocessing

Alternative 18: 35.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(x \cdot 0.5 + 1\right) - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (+ (* x 0.5) 1.0) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return ((x * 0.5) + 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 = ((x * 0.5d0) + 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 ((x * 0.5) + 1.0) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return ((x * 0.5) + 1.0) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) + 1.0) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) + 1.0) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 87.8%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 72.4%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 61.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.0%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 18.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 18.9%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. associate--l+ 20.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   5. associate-+r+ 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

7. Simplified 20.4%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in z around inf 18.7%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 18.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

10. Simplified 18.7%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

11. Taylor expanded in y around inf 13.0%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

12. Taylor expanded in x around 0 13.5%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right)} - \sqrt{x} \]
13. Step-by-step derivation

    1. *-commutative 13.5%

       \[\leadsto \left(1 + \color{blue}{x \cdot 0.5}\right) - \sqrt{x} \]

14. Simplified 13.5%

    \[\leadsto \color{blue}{\left(1 + x \cdot 0.5\right)} - \sqrt{x} \]

15. Final simplification 13.5%

    \[\leadsto \left(x \cdot 0.5 + 1\right) - \sqrt{x} \]
16. Add Preprocessing

Alternative 19: 35.0% accurate, 823.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
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;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 87.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ 87.8%

      \[\leadsto \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   3. associate-+r- 72.4%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 61.2%

      \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. associate-+r- 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} - \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

3. Simplified 44.0%

   \[\leadsto \color{blue}{\sqrt{1 + y} + \left(\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 11.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ 18.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. +-commutative 18.9%

      \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. associate--l+ 20.4%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right)\right) \]

   5. associate-+r+ 20.4%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)}\right)\right) \]

7. Simplified 20.4%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\left(\sqrt{x} + \sqrt{z}\right) + \sqrt{y}\right)\right)\right)} \]

8. Taylor expanded in z around inf 18.7%

   \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 18.7%

      \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

10. Simplified 18.7%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

11. Taylor expanded in y around inf 13.0%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

12. Taylor expanded in x around 0 32.0%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 32.0%

    \[\leadsto 1 \]
14. Add Preprocessing

Developer target: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024052 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 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))))