Main:z from

Percentage Accurate: 91.3% → 99.6%

Time: 1.3min

Alternatives: 17

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := t_1 - \sqrt{y}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;t_2 \leq 0.15:\\ \;\;\;\;\left(t_4 - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + t_3} + \frac{1}{\sqrt{y} + t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_2\right) + \left(t_3 - \sqrt{x}\right)\right) + \frac{1}{t_4 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= t_2 0.15)
     (+ (- t_4 (sqrt t)) (+ (/ 1.0 (+ (sqrt x) t_3)) (/ 1.0 (+ (sqrt y) t_1))))
     (+
      (+ (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_2) (- t_3 (sqrt x)))
      (/ 1.0 (+ t_4 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if (t_2 <= 0.15) {
		tmp = (t_4 - sqrt(t)) + ((1.0 / (sqrt(x) + t_3)) + (1.0 / (sqrt(y) + t_1)));
	} else {
		tmp = (((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x))) + (1.0 / (t_4 + sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + x))
    t_4 = sqrt((1.0d0 + t))
    if (t_2 <= 0.15d0) then
        tmp = (t_4 - sqrt(t)) + ((1.0d0 / (sqrt(x) + t_3)) + (1.0d0 / (sqrt(y) + t_1)))
    else
        tmp = (((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x))) + (1.0d0 / (t_4 + sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if (t_2 <= 0.15) {
		tmp = (t_4 - Math.sqrt(t)) + ((1.0 / (Math.sqrt(x) + t_3)) + (1.0 / (Math.sqrt(y) + t_1)));
	} else {
		tmp = (((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_2) + (t_3 - Math.sqrt(x))) + (1.0 / (t_4 + Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + x))
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if t_2 <= 0.15:
		tmp = (t_4 - math.sqrt(t)) + ((1.0 / (math.sqrt(x) + t_3)) + (1.0 / (math.sqrt(y) + t_1)))
	else:
		tmp = (((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_2) + (t_3 - math.sqrt(x))) + (1.0 / (t_4 + math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (t_2 <= 0.15)
		tmp = Float64(Float64(t_4 - sqrt(t)) + Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + Float64(1.0 / Float64(sqrt(y) + t_1))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_2) + Float64(t_3 - sqrt(x))) + Float64(1.0 / Float64(t_4 + sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + x));
	t_4 = sqrt((1.0 + t));
	tmp = 0.0;
	if (t_2 <= 0.15)
		tmp = (t_4 - sqrt(t)) + ((1.0 / (sqrt(x) + t_3)) + (1.0 / (sqrt(y) + t_1)));
	else
		tmp = (((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x))) + (1.0 / (t_4 + sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.15], N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.149999999999999994

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. +-commutative 86.3%

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

      6. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. flip-- 86.3%

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

      2. add-sqr-sqrt 68.4%

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

      3. +-commutative 68.4%

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

      4. add-sqr-sqrt 86.5%

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

      5. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. associate--l+ 89.9%

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

      2. +-inverses 89.9%

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

      3. metadata-eval 89.9%

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

   8. Simplified 89.9%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 47.3%

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

      3. add-sqr-sqrt 90.3%

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

   10. Applied egg-rr 90.3%

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

       1. associate--l+ 92.4%

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

       2. +-inverses 92.4%

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

       3. metadata-eval 92.4%

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

   12. Simplified 92.4%

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

   13. Taylor expanded in z around inf 50.6%

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

       1. +-commutative 50.6%

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

   15. Simplified 50.6%

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

   if 0.149999999999999994 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   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%

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

      2. +-commutative 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.0%

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

      2. add-sqr-sqrt 69.6%

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

      3. add-sqr-sqrt 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. associate--l+ 97.7%

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

      2. +-inverses 97.7%

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

      3. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 76.3%

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

      3. add-sqr-sqrt 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. associate--l+ 99.2%

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

       2. +-inverses 99.2%

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

       3. metadata-eval 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.15:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := t_1 - \sqrt{y}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_2 \leq 0.15:\\ \;\;\;\;t_4 + \left(\frac{1}{\sqrt{x} + t_3} + \frac{1}{\sqrt{y} + t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_2\right) + \left(t_3 - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 0.15)
     (+ t_4 (+ (/ 1.0 (+ (sqrt x) t_3)) (/ 1.0 (+ (sqrt y) t_1))))
     (+
      t_4
      (+ (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_2) (- t_3 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 0.15) {
		tmp = t_4 + ((1.0 / (sqrt(x) + t_3)) + (1.0 / (sqrt(y) + t_1)));
	} else {
		tmp = t_4 + (((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + x))
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_2 <= 0.15d0) then
        tmp = t_4 + ((1.0d0 / (sqrt(x) + t_3)) + (1.0d0 / (sqrt(y) + t_1)))
    else
        tmp = t_4 + (((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 0.15) {
		tmp = t_4 + ((1.0 / (Math.sqrt(x) + t_3)) + (1.0 / (Math.sqrt(y) + t_1)));
	} else {
		tmp = t_4 + (((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_2) + (t_3 - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + x))
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 0.15:
		tmp = t_4 + ((1.0 / (math.sqrt(x) + t_3)) + (1.0 / (math.sqrt(y) + t_1)))
	else:
		tmp = t_4 + (((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_2) + (t_3 - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 0.15)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + Float64(1.0 / Float64(sqrt(y) + t_1))));
	else
		tmp = Float64(t_4 + Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_2) + Float64(t_3 - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + x));
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 0.15)
		tmp = t_4 + ((1.0 / (sqrt(x) + t_3)) + (1.0 / (sqrt(y) + t_1)));
	else
		tmp = t_4 + (((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2) + (t_3 - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.15], N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.149999999999999994

   1. Initial program 86.3%

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

      1. associate-+l+ 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. +-commutative 86.3%

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

      6. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. flip-- 86.3%

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

      2. add-sqr-sqrt 68.4%

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

      3. +-commutative 68.4%

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

      4. add-sqr-sqrt 86.5%

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

      5. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. associate--l+ 89.9%

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

      2. +-inverses 89.9%

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

      3. metadata-eval 89.9%

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

   8. Simplified 89.9%

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

      1. flip-- 89.9%

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

      2. add-sqr-sqrt 47.3%

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

      3. add-sqr-sqrt 90.3%

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

   10. Applied egg-rr 90.3%

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

       1. associate--l+ 92.4%

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

       2. +-inverses 92.4%

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

       3. metadata-eval 92.4%

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

   12. Simplified 92.4%

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

   13. Taylor expanded in z around inf 50.6%

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

       1. +-commutative 50.6%

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

   15. Simplified 50.6%

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

   if 0.149999999999999994 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   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%

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

      2. +-commutative 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.0%

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

      2. add-sqr-sqrt 69.6%

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

      3. add-sqr-sqrt 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. associate--l+ 97.7%

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

      2. +-inverses 97.7%

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

      3. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.15:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \frac{1}{\sqrt{y} + t_1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_1 - \sqrt{y} \leq 0.0005:\\ \;\;\;\;t_3 + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (/ 1.0 (+ (sqrt y) t_1)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_1 (sqrt y)) 0.0005)
     (+ t_3 (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) t_2))
     (+ t_3 (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (+ 1.0 t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = 1.0 / (sqrt(y) + t_1);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_1 - sqrt(y)) <= 0.0005) {
		tmp = t_3 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + t_2);
	} else {
		tmp = t_3 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 + 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) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = 1.0d0 / (sqrt(y) + t_1)
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_1 - sqrt(y)) <= 0.0005d0) then
        tmp = t_3 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + t_2)
    else
        tmp = t_3 + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (1.0d0 + t_2))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = 1.0 / (Math.sqrt(y) + t_1);
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_1 - Math.sqrt(y)) <= 0.0005) {
		tmp = t_3 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + t_2);
	} else {
		tmp = t_3 + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (1.0 + t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = 1.0 / (math.sqrt(y) + t_1)
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_1 - math.sqrt(y)) <= 0.0005:
		tmp = t_3 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + t_2)
	else:
		tmp = t_3 + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (1.0 + t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(1.0 / Float64(sqrt(y) + t_1))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_1 - sqrt(y)) <= 0.0005)
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + t_2));
	else
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(1.0 + t_2)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = 1.0 / (sqrt(y) + t_1);
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_1 - sqrt(y)) <= 0.0005)
		tmp = t_3 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + t_2);
	else
		tmp = t_3 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 + t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.0005], N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 5.0000000000000001e-4

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. +-commutative 86.2%

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

      4. +-commutative 86.2%

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

      5. +-commutative 86.2%

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

      6. +-commutative 86.2%

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

   3. Simplified 86.2%

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

      1. flip-- 86.3%

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

      2. add-sqr-sqrt 68.1%

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

      3. +-commutative 68.1%

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

      4. add-sqr-sqrt 86.4%

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

      5. +-commutative 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. associate--l+ 89.8%

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

      2. +-inverses 89.8%

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

      3. metadata-eval 89.8%

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

   8. Simplified 89.8%

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

      1. flip-- 89.8%

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

      2. add-sqr-sqrt 46.9%

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

      3. add-sqr-sqrt 90.2%

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

   10. Applied egg-rr 90.2%

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

       1. associate--l+ 92.3%

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

       2. +-inverses 92.3%

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

       3. metadata-eval 92.3%

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

   12. Simplified 92.3%

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

   13. Taylor expanded in z around inf 50.2%

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

       1. +-commutative 50.2%

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

   15. Simplified 50.2%

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

   if 5.0000000000000001e-4 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   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%

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

      2. +-commutative 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.0%

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

      2. add-sqr-sqrt 73.2%

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

      3. +-commutative 73.2%

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

      4. add-sqr-sqrt 97.2%

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

      5. +-commutative 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. associate--l+ 97.7%

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

      2. +-inverses 97.7%

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

      3. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

      1. flip-- 97.7%

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

      2. add-sqr-sqrt 97.7%

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

      3. add-sqr-sqrt 97.7%

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

   10. Applied egg-rr 97.7%

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

       1. associate--l+ 97.7%

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

       2. +-inverses 97.7%

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

       3. metadata-eval 97.7%

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

   12. Simplified 97.7%

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

       1. flip-- 97.0%

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

       2. add-sqr-sqrt 69.2%

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

       3. add-sqr-sqrt 97.3%

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

   14. Applied egg-rr 98.0%

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

       1. associate--l+ 97.7%

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

       2. +-inverses 97.7%

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

       3. metadata-eval 97.7%

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

   16. Simplified 98.4%

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

   17. Taylor expanded in x around 0 52.9%

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

       1. associate-+r+ 52.9%

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

       2. +-commutative 52.9%

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

       3. +-commutative 52.9%

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

       4. +-commutative 52.9%

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

   19. Simplified 52.9%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.0005:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.1%

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

   1. associate-+l+ 92.1%

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

   2. +-commutative 92.1%

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

   3. +-commutative 92.1%

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

   4. +-commutative 92.1%

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

   5. +-commutative 92.1%

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

   6. +-commutative 92.1%

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

3. Simplified 92.1%

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

   1. flip-- 92.1%

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

   2. add-sqr-sqrt 70.9%

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

   3. +-commutative 70.9%

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

   4. add-sqr-sqrt 92.2%

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

   5. +-commutative 92.2%

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

6. Applied egg-rr 92.2%

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

   1. associate--l+ 94.1%

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

   2. +-inverses 94.1%

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

   3. metadata-eval 94.1%

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

8. Simplified 94.1%

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

   1. flip-- 94.1%

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

   2. add-sqr-sqrt 74.3%

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

   3. add-sqr-sqrt 94.3%

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

10. Applied egg-rr 94.3%

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

    1. associate--l+ 95.2%

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

    2. +-inverses 95.2%

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

    3. metadata-eval 95.2%

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

12. Simplified 95.2%

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

    1. flip-- 92.2%

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

    2. add-sqr-sqrt 68.0%

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

    3. add-sqr-sqrt 92.6%

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

14. Applied egg-rr 95.7%

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

    1. associate--l+ 93.9%

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

    2. +-inverses 93.9%

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

    3. metadata-eval 93.9%

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

16. Simplified 96.9%

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

    1. expm1-log1p-u 96.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    2. expm1-udef 93.3%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)} - 1\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

18. Applied egg-rr 93.3%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)} - 1\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
19. Step-by-step derivation

    1. expm1-def 96.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    2. expm1-log1p 96.9%

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

    3. associate-+r+ 96.9%

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

    4. +-commutative 96.9%

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

    5. +-commutative 96.9%

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

    6. +-commutative 96.9%

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

20. Simplified 96.9%

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

21. Final simplification 96.9%

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

Alternative 5: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;t_2 + 3\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(t_1 - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= z 9.5e-35)
     (+ t_2 3.0)
     (if (<= z 5.4e+16)
       (- (+ 1.0 (+ t_1 (sqrt (+ 1.0 z)))) (+ (sqrt y) (sqrt z)))
       (+ t_2 (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (- t_1 (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (z <= 9.5e-35) {
		tmp = t_2 + 3.0;
	} else if (z <= 5.4e+16) {
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(y) + sqrt(z));
	} else {
		tmp = t_2 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_1 - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (z <= 9.5d-35) then
        tmp = t_2 + 3.0d0
    else if (z <= 5.4d+16) then
        tmp = (1.0d0 + (t_1 + sqrt((1.0d0 + z)))) - (sqrt(y) + sqrt(z))
    else
        tmp = t_2 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (t_1 - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (z <= 9.5e-35) {
		tmp = t_2 + 3.0;
	} else if (z <= 5.4e+16) {
		tmp = (1.0 + (t_1 + Math.sqrt((1.0 + z)))) - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = t_2 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (t_1 - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if z <= 9.5e-35:
		tmp = t_2 + 3.0
	elif z <= 5.4e+16:
		tmp = (1.0 + (t_1 + math.sqrt((1.0 + z)))) - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = t_2 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (t_1 - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (z <= 9.5e-35)
		tmp = Float64(t_2 + 3.0);
	elseif (z <= 5.4e+16)
		tmp = Float64(Float64(1.0 + Float64(t_1 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(t_1 - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (z <= 9.5e-35)
		tmp = t_2 + 3.0;
	elseif (z <= 5.4e+16)
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(y) + sqrt(z));
	else
		tmp = t_2 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_1 - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 9.5e-35], N[(t$95$2 + 3.0), $MachinePrecision], If[LessEqual[z, 5.4e+16], N[(N[(1.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 9.5000000000000003e-35

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-+r+ 97.9%

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

      3. +-commutative 97.9%

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

      4. associate-+r+ 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l- 97.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 20.3%

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

      1. associate--l+ 41.7%

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

      2. associate-+r+ 41.7%

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

      3. +-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in z around 0 41.7%

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

      1. associate-+r+ 41.7%

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

      2. +-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in y around 0 22.0%

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

       1. associate--l+ 37.3%

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

   13. Simplified 37.3%

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

   if 9.5000000000000003e-35 < z < 5.4e16

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

      2. associate-+r+ 91.4%

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

      3. +-commutative 91.4%

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

      4. associate-+r+ 91.4%

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

      5. associate-+r+ 91.4%

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

      6. +-commutative 91.4%

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

      7. associate-+l- 91.1%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in t around inf 25.9%

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

      1. associate--l+ 31.2%

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

      2. +-commutative 31.2%

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

      3. +-commutative 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in x around 0 27.5%

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

      1. +-commutative 27.5%

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

      2. associate--l+ 31.1%

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

   10. Simplified 31.1%

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

   11. Taylor expanded in x around 0 32.4%

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

   if 5.4e16 < z

   1. Initial program 87.2%

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

      1. associate-+l+ 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. +-commutative 87.2%

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

      6. +-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. flip-- 87.2%

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

      2. add-sqr-sqrt 69.3%

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

      3. +-commutative 69.3%

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

      4. add-sqr-sqrt 87.3%

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

      5. +-commutative 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. associate--l+ 90.9%

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

      2. +-inverses 90.9%

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

      3. metadata-eval 90.9%

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

   8. Simplified 90.9%

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

   9. Taylor expanded in z around inf 66.0%

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

       1. +-commutative 66.0%

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

       2. +-commutative 66.0%

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

       3. associate-+r- 90.9%

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

       4. +-commutative 90.9%

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

   11. Simplified 90.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(1 + \left(t_2 + \sqrt{1 + z}\right)\right) - \sqrt{z}\right) + \left(t_1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 5.4e+16)
     (+ (- (+ 1.0 (+ t_2 (sqrt (+ 1.0 z)))) (sqrt z)) (- t_1 (sqrt y)))
     (+
      t_1
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 5.4e+16) {
		tmp = ((1.0 + (t_2 + sqrt((1.0 + z)))) - sqrt(z)) + (t_1 - sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_2)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + y))
    if (z <= 5.4d+16) then
        tmp = ((1.0d0 + (t_2 + sqrt((1.0d0 + z)))) - sqrt(z)) + (t_1 - sqrt(y))
    else
        tmp = t_1 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + t_2)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 5.4e+16) {
		tmp = ((1.0 + (t_2 + Math.sqrt((1.0 + z)))) - Math.sqrt(z)) + (t_1 - Math.sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + t_2)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 5.4e+16:
		tmp = ((1.0 + (t_2 + math.sqrt((1.0 + z)))) - math.sqrt(z)) + (t_1 - math.sqrt(y))
	else:
		tmp = t_1 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + 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(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 5.4e+16)
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 + sqrt(Float64(1.0 + z)))) - sqrt(z)) + Float64(t_1 - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + t_2))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 5.4e+16)
		tmp = ((1.0 + (t_2 + sqrt((1.0 + z)))) - sqrt(z)) + (t_1 - sqrt(y));
	else
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + 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[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.4e+16], N[(N[(N[(1.0 + N[(t$95$2 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.4e16

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 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- 70.7%

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

      5. associate-+r- 54.3%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in x around 0 33.1%

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

   if 5.4e16 < z

   1. Initial program 87.2%

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

      1. associate-+l+ 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. +-commutative 87.2%

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

      6. +-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. flip-- 87.2%

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

      2. add-sqr-sqrt 69.3%

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

      3. +-commutative 69.3%

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

      4. add-sqr-sqrt 87.3%

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

      5. +-commutative 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. associate--l+ 90.9%

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

      2. +-inverses 90.9%

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

      3. metadata-eval 90.9%

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

   8. Simplified 90.9%

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

      1. flip-- 90.9%

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

      2. add-sqr-sqrt 72.4%

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

      3. add-sqr-sqrt 91.2%

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

   10. Applied egg-rr 91.2%

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

       1. associate--l+ 92.7%

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

       2. +-inverses 92.7%

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

       3. metadata-eval 92.7%

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

   12. Simplified 92.7%

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

   13. Taylor expanded in z around inf 92.7%

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

       1. +-commutative 92.7%

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

   15. Simplified 92.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.8% accurate, 1.3× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 6e+24)
     (+ (+ (+ 1.0 t_2) (- (sqrt (+ 1.0 z)) (sqrt z))) (- t_1 (sqrt y)))
     (+ t_1 (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (- t_2 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 6e+24) {
		tmp = ((1.0 + t_2) + (sqrt((1.0 + z)) - sqrt(z))) + (t_1 - sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_2 - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + y))
    if (z <= 6d+24) then
        tmp = ((1.0d0 + t_2) + (sqrt((1.0d0 + z)) - sqrt(z))) + (t_1 - sqrt(y))
    else
        tmp = t_1 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (t_2 - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 6e+24) {
		tmp = ((1.0 + t_2) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (t_1 - Math.sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (t_2 - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 6e+24:
		tmp = ((1.0 + t_2) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (t_1 - math.sqrt(y))
	else:
		tmp = t_1 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (t_2 - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 6e+24)
		tmp = Float64(Float64(Float64(1.0 + t_2) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(t_1 - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(t_2 - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 6e+24)
		tmp = ((1.0 + t_2) + (sqrt((1.0 + z)) - sqrt(z))) + (t_1 - sqrt(y));
	else
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_2 - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 6e+24], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.9999999999999999e24

   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. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 70.3%

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

      5. associate-+r- 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in x around 0 33.1%

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

      1. associate-+r+ 33.1%

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

      2. associate--l+ 33.1%

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

   7. Simplified 33.1%

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

   if 5.9999999999999999e24 < z

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. +-commutative 87.5%

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

      4. +-commutative 87.5%

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

      5. +-commutative 87.5%

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

      6. +-commutative 87.5%

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

   3. Simplified 87.5%

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

      1. flip-- 87.4%

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

      2. add-sqr-sqrt 69.7%

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

      3. +-commutative 69.7%

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

      4. add-sqr-sqrt 87.6%

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

      5. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate--l+ 91.3%

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

      2. +-inverses 91.3%

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

      3. metadata-eval 91.3%

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

   8. Simplified 91.3%

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

   9. Taylor expanded in z around inf 66.2%

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

       1. +-commutative 66.2%

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

       2. +-commutative 66.2%

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

       3. associate-+r- 91.3%

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

       4. +-commutative 91.3%

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

   11. Simplified 91.3%

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

4. Final simplification 60.8%

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

Alternative 8: 95.7% accurate, 1.3× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 6e+24)
     (+ (- (+ 1.0 (+ t_2 (sqrt (+ 1.0 z)))) (sqrt z)) (- t_1 (sqrt y)))
     (+ t_1 (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (- t_2 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 6e+24) {
		tmp = ((1.0 + (t_2 + sqrt((1.0 + z)))) - sqrt(z)) + (t_1 - sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_2 - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + y))
    if (z <= 6d+24) then
        tmp = ((1.0d0 + (t_2 + sqrt((1.0d0 + z)))) - sqrt(z)) + (t_1 - sqrt(y))
    else
        tmp = t_1 + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (t_2 - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 6e+24) {
		tmp = ((1.0 + (t_2 + Math.sqrt((1.0 + z)))) - Math.sqrt(z)) + (t_1 - Math.sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (t_2 - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 6e+24:
		tmp = ((1.0 + (t_2 + math.sqrt((1.0 + z)))) - math.sqrt(z)) + (t_1 - math.sqrt(y))
	else:
		tmp = t_1 + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (t_2 - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 6e+24)
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 + sqrt(Float64(1.0 + z)))) - sqrt(z)) + Float64(t_1 - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(t_2 - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 6e+24)
		tmp = ((1.0 + (t_2 + sqrt((1.0 + z)))) - sqrt(z)) + (t_1 - sqrt(y));
	else
		tmp = t_1 + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (t_2 - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 6e+24], N[(N[(N[(1.0 + N[(t$95$2 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.9999999999999999e24

   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. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 79.6%

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

      4. associate-+l- 70.3%

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

      5. associate-+r- 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in x around 0 33.1%

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

   if 5.9999999999999999e24 < z

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. +-commutative 87.5%

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

      4. +-commutative 87.5%

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

      5. +-commutative 87.5%

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

      6. +-commutative 87.5%

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

   3. Simplified 87.5%

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

      1. flip-- 87.4%

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

      2. add-sqr-sqrt 69.7%

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

      3. +-commutative 69.7%

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

      4. add-sqr-sqrt 87.6%

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

      5. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate--l+ 91.3%

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

      2. +-inverses 91.3%

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

      3. metadata-eval 91.3%

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

   8. Simplified 91.3%

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

   9. Taylor expanded in z around inf 66.2%

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

       1. +-commutative 66.2%

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

       2. +-commutative 66.2%

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

       3. associate-+r- 91.3%

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

       4. +-commutative 91.3%

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

   11. Simplified 91.3%

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

4. Final simplification 60.8%

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

Alternative 9: 89.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 8.5 \cdot 10^{-35}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 y))))
   (if (<= z 8.5e-35)
     (+ (- (sqrt (+ 1.0 t)) (sqrt t)) 3.0)
     (if (<= z 8e+14)
       (- (+ 1.0 (+ t_1 (sqrt (+ 1.0 z)))) (+ (sqrt y) (sqrt z)))
       (+ 1.0 (- t_1 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 8.5e-35) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	} else if (z <= 8e+14) {
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(y) + sqrt(z));
	} else {
		tmp = 1.0 + (t_1 - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 8.5d-35) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + 3.0d0
    else if (z <= 8d+14) then
        tmp = (1.0d0 + (t_1 + sqrt((1.0d0 + z)))) - (sqrt(y) + sqrt(z))
    else
        tmp = 1.0d0 + (t_1 - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 8.5e-35) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 8e+14) {
		tmp = (1.0 + (t_1 + Math.sqrt((1.0 + z)))) - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = 1.0 + (t_1 - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 8.5e-35:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + 3.0
	elif z <= 8e+14:
		tmp = (1.0 + (t_1 + math.sqrt((1.0 + z)))) - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = 1.0 + (t_1 - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 8.5e-35)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + 3.0);
	elseif (z <= 8e+14)
		tmp = Float64(Float64(1.0 + Float64(t_1 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(t_1 - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 8.5e-35)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	elseif (z <= 8e+14)
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(y) + sqrt(z));
	else
		tmp = 1.0 + (t_1 - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 8.5e-35], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 8e+14], N[(N[(1.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 8.5000000000000001e-35

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-+r+ 97.9%

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

      3. +-commutative 97.9%

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

      4. associate-+r+ 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l- 97.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 20.3%

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

      1. associate--l+ 41.7%

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

      2. associate-+r+ 41.7%

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

      3. +-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in z around 0 41.7%

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

      1. associate-+r+ 41.7%

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

      2. +-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in y around 0 22.0%

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

       1. associate--l+ 37.3%

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

   13. Simplified 37.3%

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

   if 8.5000000000000001e-35 < z < 8e14

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-+r+ 92.8%

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

      3. +-commutative 92.8%

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

      4. associate-+r+ 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. associate-+l- 92.5%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in t around inf 27.0%

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

      1. associate--l+ 32.3%

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

      2. +-commutative 32.3%

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

      3. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in x around 0 28.5%

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

      1. +-commutative 28.5%

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

      2. associate--l+ 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in x around 0 33.7%

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

   if 8e14 < z

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

      2. associate-+r+ 87.0%

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

      3. +-commutative 87.0%

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

      4. associate-+r+ 87.0%

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

      5. associate-+r+ 87.0%

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

      6. +-commutative 87.0%

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

      7. associate-+l- 47.6%

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

   3. Simplified 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)} \]
   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+ 20.6%

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

      2. +-commutative 20.6%

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

      3. +-commutative 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. +-commutative 30.7%

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

   10. Simplified 30.7%

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

   11. Taylor expanded in x around 0 28.2%

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

       1. associate--l+ 48.5%

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

   13. Simplified 48.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-35}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e-34)
   (+ (- (sqrt (+ 1.0 t)) (sqrt t)) 3.0)
   (if (<= z 4.4e+14)
     (+ (sqrt (+ 1.0 x)) (- (+ 1.0 (sqrt (+ 1.0 z))) (sqrt z)))
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e-34) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	} else if (z <= 4.4e+14) {
		tmp = sqrt((1.0 + x)) + ((1.0 + sqrt((1.0 + z))) - sqrt(z));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - 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) :: tmp
    if (z <= 1.2d-34) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + 3.0d0
    else if (z <= 4.4d+14) then
        tmp = sqrt((1.0d0 + x)) + ((1.0d0 + sqrt((1.0d0 + z))) - sqrt(z))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - 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 tmp;
	if (z <= 1.2e-34) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 4.4e+14) {
		tmp = Math.sqrt((1.0 + x)) + ((1.0 + Math.sqrt((1.0 + z))) - Math.sqrt(z));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e-34:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + 3.0
	elif z <= 4.4e+14:
		tmp = math.sqrt((1.0 + x)) + ((1.0 + math.sqrt((1.0 + z))) - math.sqrt(z))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.2e-34)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + 3.0);
	elseif (z <= 4.4e+14)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(1.0 + sqrt(Float64(1.0 + z))) - sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - 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)
	tmp = 0.0;
	if (z <= 1.2e-34)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	elseif (z <= 4.4e+14)
		tmp = sqrt((1.0 + x)) + ((1.0 + sqrt((1.0 + z))) - sqrt(z));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - 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_] := If[LessEqual[z, 1.2e-34], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 4.4e+14], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.19999999999999996e-34

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-+r+ 97.9%

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

      3. +-commutative 97.9%

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

      4. associate-+r+ 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l- 97.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 20.3%

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

      1. associate--l+ 41.7%

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

      2. associate-+r+ 41.7%

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

      3. +-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in z around 0 41.7%

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

      1. associate-+r+ 41.7%

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

      2. +-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in y around 0 22.0%

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

       1. associate--l+ 37.3%

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

   13. Simplified 37.3%

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

   if 1.19999999999999996e-34 < z < 4.4e14

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-+r+ 92.8%

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

      3. +-commutative 92.8%

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

      4. associate-+r+ 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. associate-+l- 92.5%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in t around inf 27.0%

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

      1. associate--l+ 32.3%

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

      2. +-commutative 32.3%

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

      3. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in x around 0 28.5%

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

      1. +-commutative 28.5%

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

      2. associate--l+ 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in y around 0 29.6%

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

   if 4.4e14 < z

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

      2. associate-+r+ 87.0%

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

      3. +-commutative 87.0%

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

      4. associate-+r+ 87.0%

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

      5. associate-+r+ 87.0%

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

      6. +-commutative 87.0%

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

      7. associate-+l- 47.6%

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

   3. Simplified 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)} \]
   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+ 20.6%

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

      2. +-commutative 20.6%

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

      3. +-commutative 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. +-commutative 30.7%

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

   10. Simplified 30.7%

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

   11. Taylor expanded in x around 0 28.2%

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

       1. associate--l+ 48.5%

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

   13. Simplified 48.5%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e-34)
   (+ (- (sqrt (+ 1.0 t)) (sqrt t)) 3.0)
   (if (<= z 4.4e+14)
     (- (+ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt (+ 1.0 z)))) (sqrt z))
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e-34) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	} else if (z <= 4.4e+14) {
		tmp = (1.0 + (sqrt((1.0 + x)) + sqrt((1.0 + z)))) - sqrt(z);
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - 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) :: tmp
    if (z <= 1.2d-34) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + 3.0d0
    else if (z <= 4.4d+14) then
        tmp = (1.0d0 + (sqrt((1.0d0 + x)) + sqrt((1.0d0 + z)))) - sqrt(z)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - 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 tmp;
	if (z <= 1.2e-34) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + 3.0;
	} else if (z <= 4.4e+14) {
		tmp = (1.0 + (Math.sqrt((1.0 + x)) + Math.sqrt((1.0 + z)))) - Math.sqrt(z);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e-34:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + 3.0
	elif z <= 4.4e+14:
		tmp = (1.0 + (math.sqrt((1.0 + x)) + math.sqrt((1.0 + z)))) - math.sqrt(z)
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.2e-34)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + 3.0);
	elseif (z <= 4.4e+14)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 + z)))) - sqrt(z));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - 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)
	tmp = 0.0;
	if (z <= 1.2e-34)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	elseif (z <= 4.4e+14)
		tmp = (1.0 + (sqrt((1.0 + x)) + sqrt((1.0 + z)))) - sqrt(z);
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - 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_] := If[LessEqual[z, 1.2e-34], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], If[LessEqual[z, 4.4e+14], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.19999999999999996e-34

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-+r+ 97.9%

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

      3. +-commutative 97.9%

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

      4. associate-+r+ 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l- 97.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 20.3%

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

      1. associate--l+ 41.7%

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

      2. associate-+r+ 41.7%

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

      3. +-commutative 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in z around 0 41.7%

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

      1. associate-+r+ 41.7%

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

      2. +-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in y around 0 22.0%

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

       1. associate--l+ 37.3%

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

   13. Simplified 37.3%

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

   if 1.19999999999999996e-34 < z < 4.4e14

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-+r+ 92.8%

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

      3. +-commutative 92.8%

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

      4. associate-+r+ 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. associate-+l- 92.5%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in t around inf 27.0%

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

      1. associate--l+ 32.3%

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

      2. +-commutative 32.3%

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

      3. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in x around 0 28.5%

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

      1. +-commutative 28.5%

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

      2. associate--l+ 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in y around 0 29.6%

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

       1. +-commutative 29.6%

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

   13. Simplified 29.6%

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

   if 4.4e14 < z

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

      2. associate-+r+ 87.0%

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

      3. +-commutative 87.0%

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

      4. associate-+r+ 87.0%

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

      5. associate-+r+ 87.0%

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

      6. +-commutative 87.0%

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

      7. associate-+l- 47.6%

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

   3. Simplified 4.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \sqrt{z}\right)} \]
   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+ 20.6%

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

      2. +-commutative 20.6%

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

      3. +-commutative 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. +-commutative 30.7%

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

   10. Simplified 30.7%

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

   11. Taylor expanded in x around 0 28.2%

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

       1. associate--l+ 48.5%

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

   13. Simplified 48.5%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (+ 1.0 x)) (sqrt x)))) (if (<= y 0.68) (+ 1.0 t_1) t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (y <= 0.68) {
		tmp = 1.0 + t_1;
	} else {
		tmp = 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((1.0d0 + x)) - sqrt(x)
    if (y <= 0.68d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = 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((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (y <= 0.68) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if y <= 0.68:
		tmp = 1.0 + t_1
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (y <= 0.68)
		tmp = Float64(1.0 + t_1);
	else
		tmp = 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((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (y <= 0.68)
		tmp = 1.0 + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.68], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 0.680000000000000049

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate-+r+ 97.0%

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

      3. +-commutative 97.0%

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

      4. associate-+r+ 97.0%

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

      5. associate-+r+ 97.0%

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

      6. +-commutative 97.0%

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

      7. associate-+l- 79.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 18.2%

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

      1. associate--l+ 22.0%

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

      2. +-commutative 22.0%

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

      3. +-commutative 22.0%

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

   7. Simplified 22.0%

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

   8. Taylor expanded in z around inf 21.8%

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

      1. +-commutative 21.8%

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

   10. Simplified 21.8%

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

   11. Taylor expanded in y around 0 21.8%

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

       1. associate--l+ 40.8%

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

   13. Simplified 40.8%

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

   if 0.680000000000000049 < y

   1. Initial program 86.6%

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

      1. +-commutative 86.6%

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

      2. associate-+r+ 86.7%

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

      3. +-commutative 86.7%

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

      4. associate-+r+ 86.7%

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

      5. associate-+r+ 86.6%

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

      6. +-commutative 86.6%

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

      7. associate-+l- 65.6%

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

   3. Simplified 8.9%

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

   5. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 21.4%

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

      2. +-commutative 21.4%

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

      3. +-commutative 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in z around inf 20.8%

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

      1. +-commutative 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in y around inf 20.4%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.46:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.46)
   (+ (- (sqrt (+ 1.0 t)) (sqrt t)) 3.0)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.46) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - 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) :: tmp
    if (z <= 0.46d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + 3.0d0
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - 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 tmp;
	if (z <= 0.46) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + 3.0;
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.46:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + 3.0
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.46)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + 3.0);
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - 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)
	tmp = 0.0;
	if (z <= 0.46)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + 3.0;
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - 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_] := If[LessEqual[z, 0.46], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.46000000000000002

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. associate-+r+ 97.7%

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

      3. +-commutative 97.7%

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

      4. associate-+r+ 97.7%

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

      5. associate-+r+ 97.7%

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

      6. +-commutative 97.7%

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

      7. associate-+l- 97.7%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in x around 0 20.1%

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

      1. associate--l+ 40.4%

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

      2. associate-+r+ 40.4%

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

      3. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in z around 0 39.6%

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

      1. associate-+r+ 39.6%

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

      2. +-commutative 39.6%

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

   10. Simplified 39.6%

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

   11. Taylor expanded in y around 0 21.0%

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

       1. associate--l+ 36.2%

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

   13. Simplified 36.2%

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

   if 0.46000000000000002 < z

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. +-commutative 86.8%

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

      4. associate-+r+ 86.8%

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

      5. associate-+r+ 86.8%

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

      6. +-commutative 86.8%

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

      7. associate-+l- 49.1%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in t around inf 4.3%

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

      1. associate--l+ 21.4%

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

      2. +-commutative 21.4%

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

      3. +-commutative 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in z around inf 30.8%

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

      1. +-commutative 30.8%

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

   10. Simplified 30.8%

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

   11. Taylor expanded in x around 0 28.0%

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

       1. associate--l+ 47.9%

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

   13. Simplified 47.9%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.46:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. +-commutative 92.1%

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

   2. associate-+r+ 92.1%

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

   3. +-commutative 92.1%

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

   4. associate-+r+ 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l- 72.7%

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

3. Simplified 31.0%

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

5. Taylor expanded in t around inf 11.4%

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

   1. associate--l+ 21.7%

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

   2. +-commutative 21.7%

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

   3. +-commutative 21.7%

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

7. Simplified 21.7%

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

8. Taylor expanded in z around inf 21.3%

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

   1. +-commutative 21.3%

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

10. Simplified 21.3%

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

11. Taylor expanded in x around 0 26.4%

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

    1. associate--l+ 44.4%

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

13. Simplified 44.4%

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

14. Final simplification 44.4%

    \[\leadsto 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \]
15. Add Preprocessing

Alternative 15: 35.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. +-commutative 92.1%

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

   2. associate-+r+ 92.1%

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

   3. +-commutative 92.1%

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

   4. associate-+r+ 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l- 72.7%

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

3. Simplified 31.0%

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

5. Taylor expanded in t around inf 11.4%

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

   1. associate--l+ 21.7%

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

   2. +-commutative 21.7%

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

   3. +-commutative 21.7%

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

7. Simplified 21.7%

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

8. Taylor expanded in z around inf 21.3%

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

   1. +-commutative 21.3%

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

10. Simplified 21.3%

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

11. Taylor expanded in y around inf 15.2%

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

12. Final simplification 15.2%

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

Alternative 16: 35.1% accurate, 7.7× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (1.0 + (x * 0.5)) - sqrt(x);
}

Fortran

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

Java

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

Python

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 + (x * 0.5)) - sqrt(x);
end

Wolfram

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

Derivation

1. Initial program 92.1%

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

   1. +-commutative 92.1%

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

   2. associate-+r+ 92.1%

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

   3. +-commutative 92.1%

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

   4. associate-+r+ 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l- 72.7%

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

3. Simplified 31.0%

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

5. Taylor expanded in t around inf 11.4%

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

   1. associate--l+ 21.7%

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

   2. +-commutative 21.7%

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

   3. +-commutative 21.7%

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

7. Simplified 21.7%

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

8. Taylor expanded in z around inf 21.3%

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

   1. +-commutative 21.3%

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

10. Simplified 21.3%

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

11. Taylor expanded in y around inf 15.2%

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

12. Taylor expanded in x around 0 15.9%

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

    1. *-commutative 15.9%

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

14. Simplified 15.9%

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

15. Final simplification 15.9%

    \[\leadsto \left(1 + x \cdot 0.5\right) - \sqrt{x} \]
16. Add Preprocessing

Alternative 17: 34.6% 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 92.1%

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

   1. +-commutative 92.1%

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

   2. associate-+r+ 92.1%

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

   3. +-commutative 92.1%

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

   4. associate-+r+ 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l- 72.7%

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

3. Simplified 31.0%

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

5. Taylor expanded in t around inf 11.4%

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

   1. associate--l+ 21.7%

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

   2. +-commutative 21.7%

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

   3. +-commutative 21.7%

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

7. Simplified 21.7%

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

8. Taylor expanded in z around inf 21.3%

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

   1. +-commutative 21.3%

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

10. Simplified 21.3%

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

11. Taylor expanded in y around inf 15.2%

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

12. Taylor expanded in x around 0 36.4%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 36.4%

    \[\leadsto 1 \]
14. Add Preprocessing

Developer target: 99.3% 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 2024011 
(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))))