Main:z from

Percentage Accurate: 91.5% → 99.5%

Time: 38.9s

Alternatives: 25

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{x} + t\_3\\ t_5 := \sqrt{y} + t\_1\\ \mathbf{if}\;z \leq 62000000:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right)\right) + \frac{\left(1 + t\right) - t}{t\_2 + \sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 - \sqrt{t}\right) + \left(\frac{t\_5 + t\_4}{t\_5 \cdot t\_4} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 t)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (+ (sqrt x) t_3))
        (t_5 (+ (sqrt y) t_1)))
   (if (<= z 62000000.0)
     (+
      (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ (- t_1 (sqrt y)) (- t_3 (sqrt x))))
      (/ (- (+ 1.0 t) t) (+ t_2 (sqrt t))))
     (+
      (- t_2 (sqrt t))
      (+ (/ (+ t_5 t_4) (* t_5 t_4)) (* 0.5 (sqrt (/ 1.0 z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + t));
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt(x) + t_3;
	double t_5 = sqrt(y) + t_1;
	double tmp;
	if (z <= 62000000.0) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + ((t_1 - sqrt(y)) + (t_3 - sqrt(x)))) + (((1.0 + t) - t) / (t_2 + sqrt(t)));
	} else {
		tmp = (t_2 - sqrt(t)) + (((t_5 + t_4) / (t_5 * t_4)) + (0.5 * sqrt((1.0 / z))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((1.0d0 + t))
    t_3 = sqrt((1.0d0 + x))
    t_4 = sqrt(x) + t_3
    t_5 = sqrt(y) + t_1
    if (z <= 62000000.0d0) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + ((t_1 - sqrt(y)) + (t_3 - sqrt(x)))) + (((1.0d0 + t) - t) / (t_2 + sqrt(t)))
    else
        tmp = (t_2 - sqrt(t)) + (((t_5 + t_4) / (t_5 * t_4)) + (0.5d0 * sqrt((1.0d0 / z))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((1.0 + t));
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = Math.sqrt(x) + t_3;
	double t_5 = Math.sqrt(y) + t_1;
	double tmp;
	if (z <= 62000000.0) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((t_1 - Math.sqrt(y)) + (t_3 - Math.sqrt(x)))) + (((1.0 + t) - t) / (t_2 + Math.sqrt(t)));
	} else {
		tmp = (t_2 - Math.sqrt(t)) + (((t_5 + t_4) / (t_5 * t_4)) + (0.5 * Math.sqrt((1.0 / z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((1.0 + t))
	t_3 = math.sqrt((1.0 + x))
	t_4 = math.sqrt(x) + t_3
	t_5 = math.sqrt(y) + t_1
	tmp = 0
	if z <= 62000000.0:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + ((t_1 - math.sqrt(y)) + (t_3 - math.sqrt(x)))) + (((1.0 + t) - t) / (t_2 + math.sqrt(t)))
	else:
		tmp = (t_2 - math.sqrt(t)) + (((t_5 + t_4) / (t_5 * t_4)) + (0.5 * math.sqrt((1.0 / z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + t))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(sqrt(x) + t_3)
	t_5 = Float64(sqrt(y) + t_1)
	tmp = 0.0
	if (z <= 62000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) + Float64(t_3 - sqrt(x)))) + Float64(Float64(Float64(1.0 + t) - t) / Float64(t_2 + sqrt(t))));
	else
		tmp = Float64(Float64(t_2 - sqrt(t)) + Float64(Float64(Float64(t_5 + t_4) / Float64(t_5 * t_4)) + Float64(0.5 * sqrt(Float64(1.0 / z)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((1.0 + t));
	t_3 = sqrt((1.0 + x));
	t_4 = sqrt(x) + t_3;
	t_5 = sqrt(y) + t_1;
	tmp = 0.0;
	if (z <= 62000000.0)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + ((t_1 - sqrt(y)) + (t_3 - sqrt(x)))) + (((1.0 + t) - t) / (t_2 + sqrt(t)));
	else
		tmp = (t_2 - sqrt(t)) + (((t_5 + t_4) / (t_5 * t_4)) + (0.5 * sqrt((1.0 / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[z, 62000000.0], N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$5 + t$95$4), $MachinePrecision] / N[(t$95$5 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 6.2e7

   1. Initial program 96.8%

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

      1. flip-- 96.8%

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

      2. add-sqr-sqrt 77.8%

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

      3. add-sqr-sqrt 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 6.2e7 < z

   1. Initial program 83.8%

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

      1. +-commutative 83.8%

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

      2. flip-- 83.8%

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

      3. flip-- 84.0%

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

      4. frac-add 84.0%

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

   4. Applied egg-rr 84.1%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in z around inf 97.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 62000000:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{1 + t} + \sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{\left(\sqrt{y} + \sqrt{y + 1}\right) + \left(\sqrt{x} + \sqrt{1 + x}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \frac{1}{\sqrt{z} + t\_2}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{y + 1}\\ t_6 := \sqrt{y} + t\_5\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right) \leq 0.99999999999995:\\ \;\;\;\;t\_4 + \left(t\_3 + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(t\_3 + \frac{1 + \left(t\_6 + \sqrt{x}\right)}{t\_6 \cdot \left(1 + \sqrt{x}\right)}\right)\\ \end{array} \end{array} \]

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 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (/ 1.0 (+ (sqrt z) t_2)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ (sqrt y) t_5)))
   (if (<=
        (+ (- t_2 (sqrt z)) (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))))
        0.99999999999995)
     (+ t_4 (+ t_3 (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) t_1)))))
     (+ t_4 (+ t_3 (/ (+ 1.0 (+ t_6 (sqrt x))) (* t_6 (+ 1.0 (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 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = 1.0 / (sqrt(z) + t_2);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((y + 1.0));
	double t_6 = sqrt(y) + t_5;
	double tmp;
	if (((t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)))) <= 0.99999999999995) {
		tmp = t_4 + (t_3 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_1))));
	} else {
		tmp = t_4 + (t_3 + ((1.0 + (t_6 + sqrt(x))) / (t_6 * (1.0 + sqrt(x)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = 1.0d0 / (sqrt(z) + t_2)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = sqrt((y + 1.0d0))
    t_6 = sqrt(y) + t_5
    if (((t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)))) <= 0.99999999999995d0) then
        tmp = t_4 + (t_3 + ((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + t_1))))
    else
        tmp = t_4 + (t_3 + ((1.0d0 + (t_6 + sqrt(x))) / (t_6 * (1.0d0 + sqrt(x)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = 1.0 / (Math.sqrt(z) + t_2);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((y + 1.0));
	double t_6 = Math.sqrt(y) + t_5;
	double tmp;
	if (((t_2 - Math.sqrt(z)) + ((t_5 - Math.sqrt(y)) + (t_1 - Math.sqrt(x)))) <= 0.99999999999995) {
		tmp = t_4 + (t_3 + ((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + t_1))));
	} else {
		tmp = t_4 + (t_3 + ((1.0 + (t_6 + Math.sqrt(x))) / (t_6 * (1.0 + 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 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = 1.0 / (math.sqrt(z) + t_2)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((y + 1.0))
	t_6 = math.sqrt(y) + t_5
	tmp = 0
	if ((t_2 - math.sqrt(z)) + ((t_5 - math.sqrt(y)) + (t_1 - math.sqrt(x)))) <= 0.99999999999995:
		tmp = t_4 + (t_3 + ((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + t_1))))
	else:
		tmp = t_4 + (t_3 + ((1.0 + (t_6 + math.sqrt(x))) / (t_6 * (1.0 + 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 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(1.0 / Float64(sqrt(z) + t_2))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(sqrt(y) + t_5)
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x)))) <= 0.99999999999995)
		tmp = Float64(t_4 + Float64(t_3 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + t_1)))));
	else
		tmp = Float64(t_4 + Float64(t_3 + Float64(Float64(1.0 + Float64(t_6 + sqrt(x))) / Float64(t_6 * Float64(1.0 + sqrt(x))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	t_3 = 1.0 / (sqrt(z) + t_2);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = sqrt((y + 1.0));
	t_6 = sqrt(y) + t_5;
	tmp = 0.0;
	if (((t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)))) <= 0.99999999999995)
		tmp = t_4 + (t_3 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_1))));
	else
		tmp = t_4 + (t_3 + ((1.0 + (t_6 + sqrt(x))) / (t_6 * (1.0 + sqrt(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.99999999999995], N[(t$95$4 + N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(t$95$3 + N[(N[(1.0 + N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.99999999999995004

   1. Initial program 61.2%

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

      1. +-commutative 61.2%

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

      2. flip-- 61.3%

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

      3. flip-- 61.7%

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

      4. frac-add 61.7%

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

   4. Applied egg-rr 61.9%

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

   6. Simplified 80.2%

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

      1. flip-- 84.6%

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

      2. div-inv 84.6%

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

      3. add-sqr-sqrt 50.6%

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

      4. +-commutative 50.6%

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

      5. add-sqr-sqrt 85.8%

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

      6. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. associate--l+ 94.5%

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

      4. +-inverses 94.5%

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

      5. metadata-eval 94.5%

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

      6. +-commutative 94.5%

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

   10. Simplified 94.5%

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

   11. Taylor expanded in y around inf 92.7%

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

   if 0.99999999999995004 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. flip-- 97.3%

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

      3. flip-- 97.4%

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

      4. frac-add 97.4%

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

   4. Applied egg-rr 97.9%

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

   6. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. div-inv 99.1%

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

      3. add-sqr-sqrt 82.9%

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

      4. +-commutative 82.9%

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

      5. add-sqr-sqrt 99.1%

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

      6. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. associate--l+ 99.2%

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

      4. +-inverses 99.2%

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

      5. metadata-eval 99.2%

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

      6. +-commutative 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in x around 0 95.5%

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

       1. *-commutative 95.5%

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

   13. Simplified 95.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.99999999999995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \frac{1 + \left(\left(\sqrt{y} + \sqrt{y + 1}\right) + \sqrt{x}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(1 + \sqrt{x}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ t_3 := \left(\sqrt{y + 1} - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\\ t_4 := \sqrt{1 + z}\\ t_5 := \frac{1}{\sqrt{z} + t\_4}\\ \mathbf{if}\;t\_1 + \left(\left(t\_4 - \sqrt{z}\right) + t\_3\right) \leq 1.0002:\\ \;\;\;\;t\_1 + \left(t\_5 + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_5 + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- t_2 (sqrt x))))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (/ 1.0 (+ (sqrt z) t_4))))
   (if (<= (+ t_1 (+ (- t_4 (sqrt z)) t_3)) 1.0002)
     (+ t_1 (+ t_5 (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) t_2)))))
     (+ t_1 (+ t_5 t_3)))))

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 + x));
	double t_3 = (sqrt((y + 1.0)) - sqrt(y)) + (t_2 - sqrt(x));
	double t_4 = sqrt((1.0 + z));
	double t_5 = 1.0 / (sqrt(z) + t_4);
	double tmp;
	if ((t_1 + ((t_4 - sqrt(z)) + t_3)) <= 1.0002) {
		tmp = t_1 + (t_5 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_2))));
	} else {
		tmp = t_1 + (t_5 + t_3);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + x))
    t_3 = (sqrt((y + 1.0d0)) - sqrt(y)) + (t_2 - sqrt(x))
    t_4 = sqrt((1.0d0 + z))
    t_5 = 1.0d0 / (sqrt(z) + t_4)
    if ((t_1 + ((t_4 - sqrt(z)) + t_3)) <= 1.0002d0) then
        tmp = t_1 + (t_5 + ((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + t_2))))
    else
        tmp = t_1 + (t_5 + t_3)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (t_2 - Math.sqrt(x));
	double t_4 = Math.sqrt((1.0 + z));
	double t_5 = 1.0 / (Math.sqrt(z) + t_4);
	double tmp;
	if ((t_1 + ((t_4 - Math.sqrt(z)) + t_3)) <= 1.0002) {
		tmp = t_1 + (t_5 + ((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + t_2))));
	} else {
		tmp = t_1 + (t_5 + t_3);
	}
	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 + x))
	t_3 = (math.sqrt((y + 1.0)) - math.sqrt(y)) + (t_2 - math.sqrt(x))
	t_4 = math.sqrt((1.0 + z))
	t_5 = 1.0 / (math.sqrt(z) + t_4)
	tmp = 0
	if (t_1 + ((t_4 - math.sqrt(z)) + t_3)) <= 1.0002:
		tmp = t_1 + (t_5 + ((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + t_2))))
	else:
		tmp = t_1 + (t_5 + t_3)
	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 + x))
	t_3 = Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(t_2 - sqrt(x)))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(1.0 / Float64(sqrt(z) + t_4))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(t_4 - sqrt(z)) + t_3)) <= 1.0002)
		tmp = Float64(t_1 + Float64(t_5 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + t_2)))));
	else
		tmp = Float64(t_1 + Float64(t_5 + t_3));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + x));
	t_3 = (sqrt((y + 1.0)) - sqrt(y)) + (t_2 - sqrt(x));
	t_4 = sqrt((1.0 + z));
	t_5 = 1.0 / (sqrt(z) + t_4);
	tmp = 0.0;
	if ((t_1 + ((t_4 - sqrt(z)) + t_3)) <= 1.0002)
		tmp = t_1 + (t_5 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + t_2))));
	else
		tmp = t_1 + (t_5 + t_3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 1.0002], N[(t$95$1 + N[(t$95$5 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$5 + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 75.3%

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

      1. +-commutative 75.3%

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

      2. flip-- 75.3%

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

      3. flip-- 75.5%

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

      4. frac-add 75.5%

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

   4. Applied egg-rr 76.0%

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

   6. Simplified 87.6%

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

      1. flip-- 90.2%

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

      2. div-inv 90.2%

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

      3. add-sqr-sqrt 60.5%

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

      4. +-commutative 60.5%

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

      5. add-sqr-sqrt 91.0%

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

      6. +-commutative 91.0%

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

   8. Applied egg-rr 91.0%

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

      1. associate-*r/ 91.0%

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

      2. *-rgt-identity 91.0%

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

      3. associate--l+ 96.4%

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

      4. +-inverses 96.4%

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

      5. metadata-eval 96.4%

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

      6. +-commutative 96.4%

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

   10. Simplified 96.4%

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

   11. Taylor expanded in y around inf 80.7%

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

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.5%

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

      1. flip-- 99.2%

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

      2. div-inv 99.2%

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

      3. add-sqr-sqrt 84.5%

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

      4. +-commutative 84.5%

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

      5. add-sqr-sqrt 99.2%

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

      6. +-commutative 99.2%

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

   4. Applied egg-rr 97.5%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. associate--l+ 99.2%

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

      4. +-inverses 99.2%

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

      5. metadata-eval 99.2%

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

      6. +-commutative 99.2%

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

   6. Simplified 97.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) \leq 1.0002:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.6× speedup

Math

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

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

Fortran

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

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if ((t_3 + (t_5 + (t_1 - Math.sqrt(x)))) <= 1.0) {
		tmp = t_4 + ((1.0 / (Math.sqrt(z) + t_2)) + (1.0 / (Math.sqrt(x) + t_1)));
	} else {
		tmp = t_4 + (t_3 + (t_5 + (1.0 - 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 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if (t_3 + (t_5 + (t_1 - math.sqrt(x)))) <= 1.0:
		tmp = t_4 + ((1.0 / (math.sqrt(z) + t_2)) + (1.0 / (math.sqrt(x) + t_1)))
	else:
		tmp = t_4 + (t_3 + (t_5 + (1.0 - 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 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (Float64(t_3 + Float64(t_5 + Float64(t_1 - sqrt(x)))) <= 1.0)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + Float64(1.0 / Float64(sqrt(x) + t_1))));
	else
		tmp = Float64(t_4 + Float64(t_3 + Float64(t_5 + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. flip-- 83.0%

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

      3. flip-- 83.2%

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

      4. frac-add 83.2%

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

   4. Applied egg-rr 83.3%

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

   6. Simplified 91.4%

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

      1. flip-- 93.1%

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

      2. div-inv 93.1%

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

      3. add-sqr-sqrt 63.6%

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

      4. +-commutative 63.6%

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

      5. add-sqr-sqrt 93.6%

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

      6. +-commutative 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.6%

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

      2. *-rgt-identity 93.6%

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

      3. associate--l+ 97.3%

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

      4. +-inverses 97.3%

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

      5. metadata-eval 97.3%

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

      6. +-commutative 97.3%

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

   10. Simplified 97.3%

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

   11. Taylor expanded in y around inf 70.1%

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

       1. +-commutative 70.1%

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

   13. Simplified 70.1%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 70.9%

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

4. Final simplification 70.5%

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

Alternative 5: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt((y + 1.0));
	double t_2 = sqrt(x) + sqrt((1.0 + x));
	return (((t_1 + t_2) / (t_1 * t_2)) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (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
    real(8) :: t_1
    real(8) :: t_2
    t_1 = sqrt(y) + sqrt((y + 1.0d0))
    t_2 = sqrt(x) + sqrt((1.0d0 + x))
    code = (((t_1 + t_2) / (t_1 * t_2)) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) + (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) {
	double t_1 = Math.sqrt(y) + Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt(x) + Math.sqrt((1.0 + x));
	return (((t_1 + t_2) / (t_1 * t_2)) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt((y + 1.0))
	t_2 = math.sqrt(x) + math.sqrt((1.0 + x))
	return (((t_1 + t_2) / (t_1 * t_2)) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) + (math.sqrt((1.0 + t)) - math.sqrt(t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(Float64(y + 1.0)))
	t_2 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	return Float64(Float64(Float64(Float64(t_1 + t_2) / Float64(t_1 * t_2)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) + 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)
	t_1 = sqrt(y) + sqrt((y + 1.0));
	t_2 = sqrt(x) + sqrt((1.0 + x));
	tmp = (((t_1 + t_2) / (t_1 * t_2)) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) + (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_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 90.5%

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

   1. +-commutative 90.5%

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

   2. flip-- 90.5%

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

   3. flip-- 90.7%

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

   4. frac-add 90.7%

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

4. Applied egg-rr 91.1%

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

6. Simplified 95.5%

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

   1. flip-- 96.4%

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

   2. div-inv 96.4%

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

   3. add-sqr-sqrt 76.9%

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

   4. +-commutative 76.9%

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

   5. add-sqr-sqrt 96.6%

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

   6. +-commutative 96.6%

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

8. Applied egg-rr 96.6%

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

   1. associate-*r/ 96.6%

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

   2. *-rgt-identity 96.6%

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

   3. associate--l+ 98.3%

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

   4. +-inverses 98.3%

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

   5. metadata-eval 98.3%

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

   6. +-commutative 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 6: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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
    t_1 = sqrt(x) + sqrt((1.0d0 + x))
    t_2 = sqrt(y) + sqrt((y + 1.0d0))
    code = (sqrt((1.0d0 + t)) - sqrt(t)) + (((t_2 + t_1) / (t_2 * t_1)) + (sqrt((1.0d0 + z)) - sqrt(z)))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt((1.0 + x))
	t_2 = math.sqrt(y) + math.sqrt((y + 1.0))
	return (math.sqrt((1.0 + t)) - math.sqrt(t)) + (((t_2 + t_1) / (t_2 * t_1)) + (math.sqrt((1.0 + z)) - math.sqrt(z)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	t_2 = Float64(sqrt(y) + sqrt(Float64(y + 1.0)))
	return Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(t_2 + t_1) / Float64(t_2 * t_1)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt((1.0 + x));
	t_2 = sqrt(y) + sqrt((y + 1.0));
	tmp = (sqrt((1.0 + t)) - sqrt(t)) + (((t_2 + t_1) / (t_2 * t_1)) + (sqrt((1.0 + z)) - sqrt(z)));
end

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. +-commutative 90.5%

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

   2. flip-- 90.5%

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

   3. flip-- 90.7%

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

   4. frac-add 90.7%

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

4. Applied egg-rr 91.1%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 7: 97.8% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 2e-7)
     (+
      t_3
      (+
       (/ 1.0 (+ (sqrt z) t_2))
       (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))))))
     (+ t_3 (+ (- t_2 (sqrt z)) (+ t_1 (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = t_3 + ((1.0 / (sqrt(z) + t_2)) + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	} else {
		tmp = t_3 + ((t_2 - sqrt(z)) + (t_1 + (1.0 - sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0)) - sqrt(y)
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 2d-7) then
        tmp = t_3 + ((1.0d0 / (sqrt(z) + t_2)) + ((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x))))))
    else
        tmp = t_3 + ((t_2 - sqrt(z)) + (t_1 + (1.0d0 - sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = t_3 + ((1.0 / (Math.sqrt(z) + t_2)) + ((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x))))));
	} else {
		tmp = t_3 + ((t_2 - Math.sqrt(z)) + (t_1 + (1.0 - Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 2e-7:
		tmp = t_3 + ((1.0 / (math.sqrt(z) + t_2)) + ((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x))))))
	else:
		tmp = t_3 + ((t_2 - math.sqrt(z)) + (t_1 + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))));
	else
		tmp = Float64(t_3 + Float64(Float64(t_2 - sqrt(z)) + Float64(t_1 + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0)) - sqrt(y);
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 2e-7)
		tmp = t_3 + ((1.0 / (sqrt(z) + t_2)) + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	else
		tmp = t_3 + ((t_2 - sqrt(z)) + (t_1 + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 1.9999999999999999e-7

   1. Initial program 83.2%

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

      1. +-commutative 83.2%

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

      2. flip-- 83.2%

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

      3. flip-- 83.4%

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

      4. frac-add 83.4%

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

   4. Applied egg-rr 83.7%

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

   6. Simplified 91.8%

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

      1. flip-- 93.5%

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

      2. div-inv 93.5%

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

      3. add-sqr-sqrt 75.8%

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

      4. +-commutative 75.8%

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

      5. add-sqr-sqrt 94.0%

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

      6. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. *-rgt-identity 94.0%

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

      3. associate--l+ 97.3%

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

      4. +-inverses 97.3%

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

      5. metadata-eval 97.3%

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

      6. +-commutative 97.3%

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

   10. Simplified 97.3%

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

   11. Taylor expanded in y around inf 97.3%

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

   if 1.9999999999999999e-7 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0 49.7%

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

4. Final simplification 72.9%

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

Alternative 8: 97.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 2e-7)
     (+
      t_3
      (+
       t_1
       (+ (* 0.5 (sqrt (/ 1.0 y))) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))))))
     (+ t_3 (+ t_1 (+ t_2 (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 2e-7) {
		tmp = t_3 + (t_1 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	} else {
		tmp = t_3 + (t_1 + (t_2 + (1.0 - sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_2 <= 2d-7) then
        tmp = t_3 + (t_1 + ((0.5d0 * sqrt((1.0d0 / y))) + (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x))))))
    else
        tmp = t_3 + (t_1 + (t_2 + (1.0d0 - sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 2e-7) {
		tmp = t_3 + (t_1 + ((0.5 * Math.sqrt((1.0 / y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x))))));
	} else {
		tmp = t_3 + (t_1 + (t_2 + (1.0 - Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 2e-7:
		tmp = t_3 + (t_1 + ((0.5 * math.sqrt((1.0 / y))) + (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x))))))
	else:
		tmp = t_3 + (t_1 + (t_2 + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 2e-7)
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(t_2 + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 2e-7)
		tmp = t_3 + (t_1 + ((0.5 * sqrt((1.0 / y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	else
		tmp = t_3 + (t_1 + (t_2 + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 1.9999999999999999e-7

   1. Initial program 83.2%

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

      1. flip-- 83.4%

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

      2. div-inv 83.4%

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

      3. add-sqr-sqrt 65.9%

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

      4. +-commutative 65.9%

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

      5. add-sqr-sqrt 83.7%

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

      6. +-commutative 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. associate--l+ 86.6%

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

      4. +-inverses 86.6%

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

      5. metadata-eval 86.6%

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

      6. +-commutative 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in y around inf 91.8%

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

   if 1.9999999999999999e-7 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0 49.7%

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

4. Final simplification 70.3%

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

Alternative 9: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t\_1 - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_1 + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))))
   (if (<= (- t_1 (sqrt z)) 0.0001)
     (+
      (sqrt (+ 1.0 x))
      (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x))))
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (- (+ t_1 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if ((t_1 - sqrt(z)) <= 0.0001) {
		tmp = sqrt((1.0 + x)) + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if ((t_1 - sqrt(z)) <= 0.0001d0) then
        tmp = sqrt((1.0d0 + x)) + ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x)))
    else
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((t_1 + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 0.0001) {
		tmp = Math.sqrt((1.0 + x)) + ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((t_1 + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 0.0001:
		tmp = math.sqrt((1.0 + x)) + ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((t_1 + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 0.0001)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_1 + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 0.0001)
		tmp = sqrt((1.0 + x)) + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	else
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.00000000000000005e-4

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

   5. Simplified 15.4%

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

   6. Taylor expanded in z around inf 30.6%

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

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 28.5%

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

      1. associate--l+ 36.8%

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

      2. +-commutative 36.8%

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

   5. Simplified 36.8%

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

   6. Taylor expanded in y around 0 21.4%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.8% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= (- t_2 (sqrt z)) 2e-7)
     (+
      (sqrt (+ 1.0 x))
      (- (+ t_1 (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x))))
     (- (+ 1.0 (+ t_1 t_2)) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if ((t_2 - sqrt(z)) <= 2e-7) {
		tmp = sqrt((1.0 + x)) + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (t_1 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    if ((t_2 - sqrt(z)) <= 2d-7) then
        tmp = sqrt((1.0d0 + x)) + ((t_1 + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (t_1 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_2 - Math.sqrt(z)) <= 2e-7) {
		tmp = Math.sqrt((1.0 + x)) + ((t_1 + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (t_1 + t_2)) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_2 - math.sqrt(z)) <= 2e-7:
		tmp = math.sqrt((1.0 + x)) + ((t_1 + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (t_1 + t_2)) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 2e-7)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_1 + t_2)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if ((t_2 - sqrt(z)) <= 2e-7)
		tmp = sqrt((1.0 + x)) + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (t_1 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.9999999999999999e-7

   1. Initial program 84.4%

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

   3. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 15.6%

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

      2. +-commutative 15.6%

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

   5. Simplified 15.6%

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

   6. Taylor expanded in z around inf 31.1%

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

   if 1.9999999999999999e-7 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 96.0%

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

   3. Taylor expanded in t around inf 14.9%

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

      1. associate--l+ 18.3%

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

      2. +-commutative 18.3%

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

   5. Simplified 18.3%

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

   6. Taylor expanded in x around 0 12.0%

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

4. Final simplification 21.2%

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

Alternative 11: 91.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1100:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_1 + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

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 z))))
   (if (<= t 1100.0)
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (- (+ t_1 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
     (+
      (+
       (- t_1 (sqrt z))
       (+
        (- (sqrt (+ y 1.0)) (sqrt y))
        (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))))
      (* 0.5 (sqrt (/ 1.0 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 + z));
	double tmp;
	if (t <= 1100.0) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	} else {
		tmp = ((t_1 - sqrt(z)) + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))))) + (0.5 * sqrt((1.0 / 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) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (t <= 1100.0d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) + ((t_1 + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z))))
    else
        tmp = ((t_1 - sqrt(z)) + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))))) + (0.5d0 * sqrt((1.0d0 / 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 + z));
	double tmp;
	if (t <= 1100.0) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((t_1 + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	} else {
		tmp = ((t_1 - Math.sqrt(z)) + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x))))) + (0.5 * Math.sqrt((1.0 / 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 + z))
	tmp = 0
	if t <= 1100.0:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) + ((t_1 + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	else:
		tmp = ((t_1 - math.sqrt(z)) + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))))) + (0.5 * math.sqrt((1.0 / 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 + z))
	tmp = 0.0
	if (t <= 1100.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_1 + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x))))) + Float64(0.5 * sqrt(Float64(1.0 / 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 + z));
	tmp = 0.0;
	if (t <= 1100.0)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z))));
	else
		tmp = ((t_1 - sqrt(z)) + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))))) + (0.5 * sqrt((1.0 / 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 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1100.0], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1100

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0 33.5%

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

      1. associate--l+ 38.6%

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

      2. +-commutative 38.6%

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

   5. Simplified 38.6%

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

   6. Taylor expanded in y around 0 21.6%

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

   if 1100 < t

   1. Initial program 83.0%

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

   3. Taylor expanded in t around inf 85.0%

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

   4. Taylor expanded in x around 0 44.5%

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]
   5. Step-by-step derivation

      1. associate--l+ 44.5%

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

   6. Simplified 44.5%

      \[\leadsto \left(\left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1100:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 26000000:\\ \;\;\;\;\left(1 + \left(t\_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(t\_1 + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= z 26000000.0)
     (- (+ 1.0 (+ t_1 (sqrt (+ 1.0 z)))) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
     (+ 1.0 (- (+ t_1 (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (z <= 26000000.0) {
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = 1.0 + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    if (z <= 26000000.0d0) then
        tmp = (1.0d0 + (t_1 + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = 1.0d0 + ((t_1 + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (z <= 26000000.0) {
		tmp = (1.0 + (t_1 + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = 1.0 + ((t_1 + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if z <= 26000000.0:
		tmp = (1.0 + (t_1 + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = 1.0 + ((t_1 + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (z <= 26000000.0)
		tmp = Float64(Float64(1.0 + Float64(t_1 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	tmp = 0.0;
	if (z <= 26000000.0)
		tmp = (1.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = 1.0 + ((t_1 + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 26000000.0], N[(N[(1.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.6e7

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 18.5%

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

      2. +-commutative 18.5%

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

   5. Simplified 18.5%

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

   6. Taylor expanded in x around 0 12.2%

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

   if 2.6e7 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

   5. Simplified 15.4%

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

   6. Taylor expanded in x around 0 27.4%

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

   7. Taylor expanded in z around inf 34.8%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 26000000:\\ \;\;\;\;\left(1 + \left(\sqrt{y + 1} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 280000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 280000.0)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+
    1.0
    (- (+ (sqrt (+ y 1.0)) (* 0.5 (sqrt (/ 1.0 z)))) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 280000.0) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = 1.0 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 280000.0d0) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = 1.0d0 + ((sqrt((y + 1.0d0)) + (0.5d0 * sqrt((1.0d0 / z)))) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 280000.0) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = 1.0 + ((Math.sqrt((y + 1.0)) + (0.5 * Math.sqrt((1.0 / z)))) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 280000.0:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = 1.0 + ((math.sqrt((y + 1.0)) + (0.5 * math.sqrt((1.0 / z)))) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 280000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(y + 1.0)) + Float64(0.5 * sqrt(Float64(1.0 / z)))) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 280000.0)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = 1.0 + ((sqrt((y + 1.0)) + (0.5 * sqrt((1.0 / z)))) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 280000.0], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8e5

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 18.5%

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

      2. +-commutative 18.5%

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

   5. Simplified 18.5%

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

   6. Taylor expanded in x around 0 21.0%

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

   7. Taylor expanded in y around 0 9.8%

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

   if 2.8e5 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

   5. Simplified 15.4%

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

   6. Taylor expanded in x around 0 27.4%

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

   7. Taylor expanded in z around inf 34.8%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 280000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 25500000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 25500000.0)
   (- (+ (sqrt (+ 1.0 z)) 2.0) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25500000.0) {
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 25500000.0d0) then
        tmp = (sqrt((1.0d0 + z)) + 2.0d0) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = sqrt((1.0d0 + x)) + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 25500000.0) {
		tmp = (Math.sqrt((1.0 + z)) + 2.0) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 25500000.0:
		tmp = (math.sqrt((1.0 + z)) + 2.0) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 25500000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) + 2.0) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 25500000.0)
		tmp = (sqrt((1.0 + z)) + 2.0) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = sqrt((1.0 + x)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 25500000.0], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.55e7

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf 15.1%

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

      1. associate--l+ 18.5%

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

      2. +-commutative 18.5%

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

   5. Simplified 18.5%

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

   6. Taylor expanded in x around 0 21.0%

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

   7. Taylor expanded in y around 0 9.8%

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

   if 2.55e7 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 3.9%

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

      1. associate--l+ 15.4%

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

      2. +-commutative 15.4%

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

   5. Simplified 15.4%

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

   6. Taylor expanded in z around inf 30.6%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 25500000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.1% accurate, 2.6× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.0)
   (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))
   (/ (+ (* -0.125 (sqrt (/ 1.0 x))) (* (sqrt x) 0.5)) x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    else
        tmp = (((-0.125d0) * sqrt((1.0d0 / x))) + (sqrt(x) * 0.5d0)) / x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = ((-0.125 * Math.sqrt((1.0 / x))) + (Math.sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = ((-0.125 * math.sqrt((1.0 / x))) + (math.sqrt(x) * 0.5)) / x
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(1.0 / x))) + Float64(sqrt(x) * 0.5)) / x);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	else
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.0], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around 0 29.4%

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

   7. Taylor expanded in z around inf 38.6%

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

   if 1 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

      \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.0%

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

Alternative 16: 40.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 110000:\\ \;\;\;\;\mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 110000.0)
   (- (hypot 1.0 (sqrt x)) (sqrt x))
   (/ (+ (* -0.125 (sqrt (/ 1.0 x))) (* (sqrt x) 0.5)) x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 110000.0) {
		tmp = hypot(1.0, sqrt(x)) - sqrt(x);
	} else {
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 110000.0) {
		tmp = Math.hypot(1.0, Math.sqrt(x)) - Math.sqrt(x);
	} else {
		tmp = ((-0.125 * Math.sqrt((1.0 / x))) + (Math.sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 110000.0:
		tmp = math.hypot(1.0, math.sqrt(x)) - math.sqrt(x)
	else:
		tmp = ((-0.125 * math.sqrt((1.0 / x))) + (math.sqrt(x) * 0.5)) / x
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 110000.0)
		tmp = Float64(hypot(1.0, sqrt(x)) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(1.0 / x))) + Float64(sqrt(x) * 0.5)) / x);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 110000.0)
		tmp = hypot(1.0, sqrt(x)) - sqrt(x);
	else
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 110000.0], N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e5

   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. Add Preprocessing

   3. Taylor expanded in t around inf 15.2%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in x around inf 23.0%

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

      1. mul-1-neg 23.0%

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

   8. Simplified 23.0%

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

      1. add-sqr-sqrt 23.0%

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

      2. hypot-1-def 23.0%

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

   10. Applied egg-rr 23.0%

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

   if 1.1e5 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 4.7%

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

      1. associate--l+ 5.9%

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

      2. +-commutative 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf 3.8%

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

      1. mul-1-neg 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in x around inf 9.5%

      \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 110000:\\ \;\;\;\;\mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 110000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 110000.0)
   (- (sqrt (+ 1.0 x)) (sqrt x))
   (/ (+ (* -0.125 (sqrt (/ 1.0 x))) (* (sqrt x) 0.5)) x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 110000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 110000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = (((-0.125d0) * sqrt((1.0d0 / x))) + (sqrt(x) * 0.5d0)) / x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 110000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = ((-0.125 * Math.sqrt((1.0 / x))) + (Math.sqrt(x) * 0.5)) / x;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 110000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = ((-0.125 * math.sqrt((1.0 / x))) + (math.sqrt(x) * 0.5)) / x
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 110000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(1.0 / x))) + Float64(sqrt(x) * 0.5)) / x);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 110000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = ((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 110000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e5

   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. Add Preprocessing

   3. Taylor expanded in t around inf 15.2%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in x around inf 23.0%

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

      1. mul-1-neg 23.0%

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

   8. Simplified 23.0%

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

      1. unsub-neg 23.0%

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

   10. Applied egg-rr 23.0%

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

   if 1.1e5 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 4.7%

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

      1. associate--l+ 5.9%

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

      2. +-commutative 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf 3.8%

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

      1. mul-1-neg 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in x around inf 9.5%

      \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 110000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 75000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 75000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 75000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 75000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 75000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 75000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 75000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 75000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 75000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e7

   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. Add Preprocessing

   3. Taylor expanded in t around inf 15.2%

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

      1. associate--l+ 29.5%

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

      2. +-commutative 29.5%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in x around inf 23.0%

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

      1. mul-1-neg 23.0%

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

   8. Simplified 23.0%

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

      1. unsub-neg 23.0%

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

   10. Applied egg-rr 23.0%

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

   if 7.5e7 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 4.7%

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

      1. associate--l+ 5.9%

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

      2. +-commutative 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf 3.8%

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

      1. mul-1-neg 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in x around inf 9.5%

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

Alternative 19: 40.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.32)
   (- (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))) (sqrt x))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.32) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.32d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.32) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.32:
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.32)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.32)
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.32], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32000000000000006

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around inf 22.6%

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

      1. mul-1-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 22.6%

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

   if 1.32000000000000006 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.05)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.05d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.05:
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.05)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.05)
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.05], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around inf 22.6%

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

      1. mul-1-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 22.6%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. associate--l+ 22.6%

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

   11. Simplified 22.6%

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

   if 1.05000000000000004 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 39.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around inf 22.6%

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

      1. mul-1-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 22.6%

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

   if 1 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

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

4. Final simplification 15.6%

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

Alternative 22: 39.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around inf 22.6%

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

      1. mul-1-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 22.6%

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

       1. associate--l+ 22.6%

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

   11. Simplified 22.6%

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

   if 1 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

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

4. Final simplification 15.6%

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

Alternative 23: 39.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.22:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 0.22) (- 1.0 (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.22) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 0.22d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.22) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 0.22:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 0.22)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 0.22)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 0.22], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.220000000000000001

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 15.4%

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

      1. associate--l+ 29.9%

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

      2. +-commutative 29.9%

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

   5. Simplified 29.9%

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

   6. Taylor expanded in x around inf 22.6%

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

      1. mul-1-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 22.6%

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

   if 0.220000000000000001 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.2%

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

      2. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around inf 4.6%

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

      1. mul-1-neg 4.6%

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

   8. Simplified 4.6%

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

   9. Taylor expanded in x around inf 9.7%

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

Alternative 24: 34.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in t around inf 9.6%

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

   1. associate--l+ 17.0%

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

   2. +-commutative 17.0%

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

5. Simplified 17.0%

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

6. Taylor expanded in x around inf 12.8%

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

   1. mul-1-neg 12.8%

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

8. Simplified 12.8%

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

9. Taylor expanded in x around 0 11.1%

   \[\leadsto \color{blue}{1 - \sqrt{x}} \]
10. Add Preprocessing

Alternative 25: 1.9% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in t around inf 9.6%

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

   1. associate--l+ 17.0%

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

   2. +-commutative 17.0%

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

5. Simplified 17.0%

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

6. Taylor expanded in x around inf 12.8%

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

   1. mul-1-neg 12.8%

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

8. Simplified 12.8%

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

9. Taylor expanded in x around 0 11.1%

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

10. Taylor expanded in x around inf 1.6%

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

    1. neg-mul-1 1.6%

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

12. Simplified 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]
13. Add Preprocessing

Developer Target 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))