Main:z from

Percentage Accurate: 92.0% → 99.7%

Time: 58.3s

Alternatives: 28

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.0% 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: 99.7% accurate, 0.8× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 5.00000000000000024e-5

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 73.4%

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

      4. +-commutative 73.4%

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

      5. add-sqr-sqrt 83.4%

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

      6. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate--l+ 86.0%

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

      2. +-inverses 86.0%

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

      3. metadata-eval 86.0%

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

      4. *-lft-identity 86.0%

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

      5. +-commutative 86.0%

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

   8. Simplified 86.0%

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

      1. flip-- 86.1%

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

      2. div-inv 86.1%

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

      3. add-sqr-sqrt 74.1%

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

      4. add-sqr-sqrt 86.5%

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

      5. associate--l+ 90.4%

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

   10. Applied egg-rr 90.4%

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

       1. +-inverses 90.4%

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

       2. metadata-eval 90.4%

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

       3. *-lft-identity 90.4%

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

       4. +-commutative 90.4%

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

   12. Simplified 90.4%

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

   13. Taylor expanded in t around inf 48.8%

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

   14. Taylor expanded in z around inf 53.7%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.4%

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

      2. div-inv 97.4%

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

      3. add-sqr-sqrt 78.5%

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

      4. +-commutative 78.5%

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

      5. add-sqr-sqrt 97.9%

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

      6. associate--l+ 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. associate-*r/ 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate-+r- 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r- 98.1%

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

      6. +-inverses 98.1%

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

      7. metadata-eval 98.1%

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

      8. +-commutative 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 75.7%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 5.00000000000000024e-5

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 73.4%

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

      4. +-commutative 73.4%

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

      5. add-sqr-sqrt 83.4%

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

      6. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate--l+ 86.0%

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

      2. +-inverses 86.0%

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

      3. metadata-eval 86.0%

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

      4. *-lft-identity 86.0%

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

      5. +-commutative 86.0%

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

   8. Simplified 86.0%

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

      1. flip-- 86.1%

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

      2. div-inv 86.1%

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

      3. add-sqr-sqrt 74.1%

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

      4. add-sqr-sqrt 86.5%

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

      5. associate--l+ 90.4%

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

   10. Applied egg-rr 90.4%

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

       1. +-inverses 90.4%

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

       2. metadata-eval 90.4%

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

       3. *-lft-identity 90.4%

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

       4. +-commutative 90.4%

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

   12. Simplified 90.4%

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

   13. Taylor expanded in t around inf 48.8%

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

   14. Taylor expanded in z around inf 53.7%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 39.6%

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

      1. flip-- 97.4%

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

      2. div-inv 97.4%

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

      3. add-sqr-sqrt 78.5%

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

      4. +-commutative 78.5%

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

      5. add-sqr-sqrt 97.9%

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

      6. associate--l+ 97.9%

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

   7. Applied egg-rr 39.7%

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

      1. associate-*r/ 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate-+r- 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r- 98.1%

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

      6. +-inverses 98.1%

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

      7. metadata-eval 98.1%

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

      8. +-commutative 98.1%

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

   9. Simplified 39.7%

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

4. Final simplification 46.7%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 70000000:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t\_1}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 70000000.0)
     (+
      (+ (- t_1 (sqrt y)) (- 1.0 (sqrt x)))
      (+ (- (sqrt (+ z 1.0)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) t_1)))
      (* 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((1.0 + y));
	double tmp;
	if (z <= 70000000.0) {
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + ((sqrt((z + 1.0)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t)));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 70000000.0d0) then
        tmp = ((t_1 - sqrt(y)) + (1.0d0 - sqrt(x))) + ((sqrt((z + 1.0d0)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t)))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + t_1))) + (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((1.0 + y));
	double tmp;
	if (z <= 70000000.0) {
		tmp = ((t_1 - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + t_1))) + (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((1.0 + y))
	tmp = 0
	if z <= 70000000.0:
		tmp = ((t_1 - math.sqrt(y)) + (1.0 - math.sqrt(x))) + ((math.sqrt((z + 1.0)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t)))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + t_1))) + (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(1.0 + y))
	tmp = 0.0
	if (z <= 70000000.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 - sqrt(x))) + Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + t_1))) + 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((1.0 + y));
	tmp = 0.0;
	if (z <= 70000000.0)
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + ((sqrt((z + 1.0)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t)));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 70000000.0], N[(N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 7e7

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 39.6%

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

   if 7e7 < z

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 73.4%

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

      4. +-commutative 73.4%

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

      5. add-sqr-sqrt 83.4%

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

      6. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate--l+ 86.0%

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

      2. +-inverses 86.0%

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

      3. metadata-eval 86.0%

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

      4. *-lft-identity 86.0%

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

      5. +-commutative 86.0%

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

   8. Simplified 86.0%

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

      1. flip-- 86.1%

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

      2. div-inv 86.1%

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

      3. add-sqr-sqrt 74.1%

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

      4. add-sqr-sqrt 86.5%

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

      5. associate--l+ 90.4%

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

   10. Applied egg-rr 90.4%

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

       1. +-inverses 90.4%

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

       2. metadata-eval 90.4%

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

       3. *-lft-identity 90.4%

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

       4. +-commutative 90.4%

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

   12. Simplified 90.4%

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

   13. Taylor expanded in t around inf 48.8%

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

   14. Taylor expanded in z around inf 53.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 70000000:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2100000:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

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 2100000.0)
   (+
    (+ (- (sqrt (+ z 1.0)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t)))
    (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))))
   (+
    (+
     (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
     (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
    (* 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 tmp;
	if (z <= 2100000.0) {
		tmp = ((sqrt((z + 1.0)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (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) :: tmp
    if (z <= 2100000.0d0) then
        tmp = ((sqrt((z + 1.0d0)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y)))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + sqrt((1.0d0 + y))))) + (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 tmp;
	if (z <= 2100000.0) {
		tmp = ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y)));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y))))) + (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):
	tmp = 0
	if z <= 2100000.0:
		tmp = ((math.sqrt((z + 1.0)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y)))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + math.sqrt((1.0 + y))))) + (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)
	tmp = 0.0
	if (z <= 2100000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) + 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)
	tmp = 0.0;
	if (z <= 2100000.0)
		tmp = ((sqrt((z + 1.0)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (1.0 - sqrt(y)));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (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_] := If[LessEqual[z, 2100000.0], N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.1e6

   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. Step-by-step derivation

      1. associate-+l+ 97.5%

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

      2. sub-neg 97.5%

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

      3. sub-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in x around 0 40.5%

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

   6. Taylor expanded in y around 0 19.8%

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

   if 2.1e6 < z

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 73.1%

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

      4. +-commutative 73.1%

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

      5. add-sqr-sqrt 83.4%

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

      6. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate--l+ 85.9%

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

      2. +-inverses 85.9%

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

      3. metadata-eval 85.9%

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

      4. *-lft-identity 85.9%

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

      5. +-commutative 85.9%

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

   8. Simplified 85.9%

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

      1. flip-- 86.0%

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

      2. div-inv 86.0%

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

      3. add-sqr-sqrt 74.2%

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

      4. add-sqr-sqrt 86.4%

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

      5. associate--l+ 90.2%

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

   10. Applied egg-rr 90.2%

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

       1. +-inverses 90.2%

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

       2. metadata-eval 90.2%

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

       3. *-lft-identity 90.2%

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

       4. +-commutative 90.2%

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

   12. Simplified 90.2%

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

   13. Taylor expanded in t around inf 48.4%

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

   14. Taylor expanded in z around inf 53.2%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2100000:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1200000000:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + t\_1}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 1200000000.0)
     (+
      (+ (- t_1 (sqrt y)) (- 1.0 (sqrt x)))
      (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (/ 1.0 (+ (sqrt y) t_1)))
      (* 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((1.0 + y));
	double tmp;
	if (z <= 1200000000.0) {
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 1200000000.0d0) then
        tmp = ((t_1 - sqrt(y)) + (1.0d0 - sqrt(x))) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (1.0d0 / (sqrt(y) + t_1))) + (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((1.0 + y));
	double tmp;
	if (z <= 1200000000.0) {
		tmp = ((t_1 - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (1.0 / (Math.sqrt(y) + t_1))) + (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((1.0 + y))
	tmp = 0
	if z <= 1200000000.0:
		tmp = ((t_1 - math.sqrt(y)) + (1.0 - math.sqrt(x))) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (1.0 / (math.sqrt(y) + t_1))) + (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(1.0 + y))
	tmp = 0.0
	if (z <= 1200000000.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 - sqrt(x))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(1.0 / Float64(sqrt(y) + t_1))) + 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((1.0 + y));
	tmp = 0.0;
	if (z <= 1200000000.0)
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (1.0 / (sqrt(y) + t_1))) + (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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1200000000.0], N[(N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e9

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. sub-neg 96.9%

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

      3. sub-neg 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

      6. +-commutative 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 24.8%

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

      1. flip-- 24.9%

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

      2. div-inv 24.9%

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

      3. add-sqr-sqrt 25.0%

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

      4. add-sqr-sqrt 25.0%

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

      5. associate--l+ 25.0%

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

   8. Applied egg-rr 25.0%

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

      1. +-inverses 25.0%

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

      2. metadata-eval 25.0%

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

      3. *-lft-identity 25.0%

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

      4. +-commutative 25.0%

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

   10. Simplified 25.0%

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

   if 1.2e9 < z

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 73.2%

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

      4. +-commutative 73.2%

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

      5. add-sqr-sqrt 83.4%

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

      6. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate--l+ 86.1%

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

      2. +-inverses 86.1%

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

      3. metadata-eval 86.1%

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

      4. *-lft-identity 86.1%

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

      5. +-commutative 86.1%

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

   8. Simplified 86.1%

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

      1. flip-- 86.1%

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

      2. div-inv 86.1%

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

      3. add-sqr-sqrt 73.9%

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

      4. add-sqr-sqrt 86.5%

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

      5. associate--l+ 90.5%

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

   10. Applied egg-rr 90.5%

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

       1. +-inverses 90.5%

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

       2. metadata-eval 90.5%

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

       3. *-lft-identity 90.5%

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

       4. +-commutative 90.5%

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

   12. Simplified 90.5%

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

   13. Taylor expanded in t around inf 48.3%

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

   14. Taylor expanded in z around inf 53.1%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1200000000:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ \mathbf{if}\;y \leq 1450000:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{z}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot {y}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))))
   (if (<= y 1450000.0)
     (+
      (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- 1.0 (sqrt x)))
      (/ 1.0 (+ (sqrt z) t_1)))
     (+
      (- t_1 (sqrt z))
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 (pow y -0.5)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double tmp;
	if (y <= 1450000.0) {
		tmp = ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + t_1));
	} else {
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * pow(y, -0.5)));
	}
	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((z + 1.0d0))
    if (y <= 1450000.0d0) then
        tmp = ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 - sqrt(x))) + (1.0d0 / (sqrt(z) + t_1))
    else
        tmp = (t_1 - sqrt(z)) + ((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * (y ** (-0.5d0))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double tmp;
	if (y <= 1450000.0) {
		tmp = ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + (1.0 / (Math.sqrt(z) + t_1));
	} else {
		tmp = (t_1 - Math.sqrt(z)) + ((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * Math.pow(y, -0.5)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	tmp = 0
	if y <= 1450000.0:
		tmp = ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (1.0 - math.sqrt(x))) + (1.0 / (math.sqrt(z) + t_1))
	else:
		tmp = (t_1 - math.sqrt(z)) + ((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * math.pow(y, -0.5)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (y <= 1450000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 - sqrt(x))) + Float64(1.0 / Float64(sqrt(z) + t_1)));
	else
		tmp = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(0.5 * (y ^ -0.5))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	tmp = 0.0;
	if (y <= 1450000.0)
		tmp = ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + t_1));
	else
		tmp = (t_1 - sqrt(z)) + ((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * (y ^ -0.5)));
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1450000.0], N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Power[y, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e6

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 47.7%

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

   6. Taylor expanded in t around inf 27.7%

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

      1. flip-- 27.8%

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

      2. div-inv 27.8%

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

      3. add-sqr-sqrt 18.9%

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

      4. add-sqr-sqrt 28.0%

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

      5. associate--l+ 28.2%

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

   8. Applied egg-rr 28.2%

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

      1. +-inverses 28.2%

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

      2. metadata-eval 28.2%

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

      3. *-lft-identity 28.2%

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

      4. +-commutative 28.2%

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

   10. Simplified 28.2%

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

   if 1.45e6 < y

   1. Initial program 82.6%

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

      1. associate-+l+ 82.6%

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

      2. sub-neg 82.6%

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

      3. sub-neg 82.6%

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

      4. +-commutative 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

   3. Simplified 82.6%

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

      1. flip-- 82.7%

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

      2. div-inv 82.7%

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

      3. add-sqr-sqrt 64.3%

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

      4. +-commutative 64.3%

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

      5. add-sqr-sqrt 83.3%

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

      6. +-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate--l+ 86.0%

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

      2. +-inverses 86.0%

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

      3. metadata-eval 86.0%

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

      4. *-lft-identity 86.0%

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

      5. +-commutative 86.0%

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

   8. Simplified 86.0%

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

      1. flip-- 86.1%

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

      2. div-inv 86.1%

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

      3. add-sqr-sqrt 51.0%

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

      4. add-sqr-sqrt 86.6%

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

      5. associate--l+ 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. +-inverses 91.1%

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

       2. metadata-eval 91.1%

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

       3. *-lft-identity 91.1%

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

       4. +-commutative 91.1%

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

   12. Simplified 91.1%

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

   13. Taylor expanded in t around inf 48.6%

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

   14. Taylor expanded in y around inf 48.5%

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

       1. rem-exp-log 48.4%

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

       2. exp-neg 48.4%

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

       3. unpow1/2 48.4%

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

       4. exp-prod 48.3%

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

       5. distribute-lft-neg-out 48.3%

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

       6. distribute-rgt-neg-in 48.3%

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

       7. metadata-eval 48.3%

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

       8. exp-to-pow 48.5%

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

   16. Simplified 48.5%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1450000:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot {y}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 4.5 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

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 y))))
   (if (<= z 4.5e+26)
     (+
      (+ (- t_1 (sqrt y)) (- 1.0 (sqrt x)))
      (/ 1.0 (+ (sqrt z) (sqrt (+ z 1.0)))))
     (+
      (+ (/ 1.0 (+ (sqrt y) t_1)) (+ 1.0 (- (* x 0.5) (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 + y));
	double tmp;
	if (z <= 4.5e+26) {
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	} else {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
    if (z <= 4.5d+26) then
        tmp = ((t_1 - sqrt(y)) + (1.0d0 - sqrt(x))) + (1.0d0 / (sqrt(z) + sqrt((z + 1.0d0))))
    else
        tmp = ((1.0d0 / (sqrt(y) + t_1)) + (1.0d0 + ((x * 0.5d0) - 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 + y));
	double tmp;
	if (z <= 4.5e+26) {
		tmp = ((t_1 - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + (1.0 / (Math.sqrt(z) + Math.sqrt((z + 1.0))));
	} else {
		tmp = ((1.0 / (Math.sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
	tmp = 0
	if z <= 4.5e+26:
		tmp = ((t_1 - math.sqrt(y)) + (1.0 - math.sqrt(x))) + (1.0 / (math.sqrt(z) + math.sqrt((z + 1.0))))
	else:
		tmp = ((1.0 / (math.sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
	tmp = 0.0
	if (z <= 4.5e+26)
		tmp = Float64(Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 - sqrt(x))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(z + 1.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 + Float64(Float64(x * 0.5) - 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 + y));
	tmp = 0.0;
	if (z <= 4.5e+26)
		tmp = ((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + sqrt((z + 1.0))));
	else
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 4.5e+26], N[(N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $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 z < 4.49999999999999978e26

   1. Initial program 94.2%

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

      1. associate-+l+ 94.2%

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

      2. sub-neg 94.2%

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

      3. sub-neg 94.2%

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

      4. +-commutative 94.2%

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

      5. +-commutative 94.2%

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

      6. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in t around inf 24.2%

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

      1. flip-- 24.3%

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

      2. div-inv 24.3%

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

      3. add-sqr-sqrt 24.1%

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

      4. add-sqr-sqrt 24.4%

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

      5. associate--l+ 24.7%

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

   8. Applied egg-rr 24.7%

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

      1. +-inverses 24.7%

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

      2. metadata-eval 24.7%

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

      3. *-lft-identity 24.7%

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

      4. +-commutative 24.7%

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

   10. Simplified 24.7%

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

   if 4.49999999999999978e26 < z

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. sub-neg 84.7%

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

      3. sub-neg 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. +-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in t around inf 40.6%

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

   6. Taylor expanded in x around 0 28.4%

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

      1. associate--l+ 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in z around inf 28.4%

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

       1. flip-- 88.2%

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

       2. div-inv 88.2%

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

       3. add-sqr-sqrt 75.5%

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

       4. add-sqr-sqrt 88.7%

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

       5. associate--l+ 93.1%

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

   11. Applied egg-rr 28.5%

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

       1. +-inverses 93.1%

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

       2. metadata-eval 93.1%

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

       3. *-lft-identity 93.1%

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

       4. +-commutative 93.1%

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

   13. Simplified 28.5%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.95 \cdot 10^{+25}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

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 y))))
   (if (<= z 1.95e+25)
     (+ (- (sqrt (+ z 1.0)) (sqrt z)) (+ 1.0 (- t_1 (+ (sqrt x) (sqrt y)))))
     (+
      (+ (/ 1.0 (+ (sqrt y) t_1)) (+ 1.0 (- (* x 0.5) (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 + y));
	double tmp;
	if (z <= 1.95e+25) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	} else {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
    if (z <= 1.95d+25) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + (1.0d0 + (t_1 - (sqrt(x) + sqrt(y))))
    else
        tmp = ((1.0d0 / (sqrt(y) + t_1)) + (1.0d0 + ((x * 0.5d0) - 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 + y));
	double tmp;
	if (z <= 1.95e+25) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (1.0 + (t_1 - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = ((1.0 / (Math.sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
	tmp = 0
	if z <= 1.95e+25:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + (1.0 + (t_1 - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = ((1.0 / (math.sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y))
	tmp = 0.0
	if (z <= 1.95e+25)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 + Float64(Float64(x * 0.5) - 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 + y));
	tmp = 0.0;
	if (z <= 1.95e+25)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (1.0 + (t_1 - (sqrt(x) + sqrt(y))));
	else
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 + ((x * 0.5) - 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 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.95e+25], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $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 z < 1.9500000000000001e25

   1. Initial program 94.3%

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

      1. associate-+l+ 94.3%

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

      2. sub-neg 94.3%

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

      3. sub-neg 94.3%

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

      4. +-commutative 94.3%

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

      5. +-commutative 94.3%

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

      6. +-commutative 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in x around 0 38.6%

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

   6. Taylor expanded in t around inf 24.2%

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

   7. Taylor expanded in x around 0 19.4%

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

      1. associate--l+ 31.7%

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

      2. +-commutative 31.7%

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

   9. Simplified 31.7%

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

   if 1.9500000000000001e25 < z

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. sub-neg 84.7%

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

      3. sub-neg 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. +-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in t around inf 40.3%

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

   6. Taylor expanded in x around 0 28.2%

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

      1. associate--l+ 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in z around inf 28.2%

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

       1. flip-- 88.2%

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

       2. div-inv 88.2%

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

       3. add-sqr-sqrt 75.6%

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

       4. add-sqr-sqrt 88.7%

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

       5. associate--l+ 93.0%

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

   11. Applied egg-rr 28.3%

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

       1. +-inverses 93.0%

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

       2. metadata-eval 93.0%

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

       3. *-lft-identity 93.0%

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

       4. +-commutative 93.0%

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

   13. Simplified 28.3%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+25}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.5e-7)
   (+
    (- (sqrt (+ z 1.0)) (sqrt z))
    (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y)))))
   (+
    (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (+ 1.0 (- (* x 0.5) (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 tmp;
	if (y <= 4.5e-7) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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) :: tmp
    if (y <= 4.5d-7) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + (2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y))))
    else
        tmp = ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (1.0d0 + ((x * 0.5d0) - 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 tmp;
	if (y <= 4.5e-7) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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):
	tmp = 0
	if y <= 4.5e-7:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + (2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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)
	tmp = 0.0
	if (y <= 4.5e-7)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(1.0 + Float64(Float64(x * 0.5) - 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)
	tmp = 0.0;
	if (y <= 4.5e-7)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	else
		tmp = ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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_] := If[LessEqual[y, 4.5e-7], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $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 y < 4.4999999999999998e-7

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 46.8%

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

   6. Taylor expanded in t around inf 26.6%

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

   7. Taylor expanded in y around 0 26.7%

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

      1. associate--l+ 26.7%

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

      2. *-commutative 26.7%

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

      3. +-commutative 26.7%

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

   9. Simplified 26.7%

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

   if 4.4999999999999998e-7 < y

   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. Step-by-step derivation

      1. associate-+l+ 83.0%

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

      2. sub-neg 83.0%

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

      3. sub-neg 83.0%

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

      4. +-commutative 83.0%

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

      5. +-commutative 83.0%

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

      6. +-commutative 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t around inf 40.5%

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

   6. Taylor expanded in x around 0 25.8%

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

      1. associate--l+ 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in z around inf 16.0%

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

       1. flip-- 86.4%

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

       2. div-inv 86.4%

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

       3. add-sqr-sqrt 52.1%

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

       4. add-sqr-sqrt 86.9%

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

       5. associate--l+ 91.3%

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

   11. Applied egg-rr 16.1%

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

       1. +-inverses 91.3%

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

       2. metadata-eval 91.3%

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

       3. *-lft-identity 91.3%

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

       4. +-commutative 91.3%

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

   13. Simplified 16.1%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00047:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 + \left(y \cdot \left(0.5 + y \cdot -0.125\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.00047)
   (+
    (- (sqrt (+ z 1.0)) (sqrt z))
    (+ (- 1.0 (sqrt x)) (+ 1.0 (- (* y (+ 0.5 (* y -0.125))) (sqrt y)))))
   (+
    (+ (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))) (+ 1.0 (- (* x 0.5) (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 tmp;
	if (y <= 0.00047) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((1.0 - sqrt(x)) + (1.0 + ((y * (0.5 + (y * -0.125))) - sqrt(y))));
	} else {
		tmp = ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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) :: tmp
    if (y <= 0.00047d0) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + ((1.0d0 - sqrt(x)) + (1.0d0 + ((y * (0.5d0 + (y * (-0.125d0)))) - sqrt(y))))
    else
        tmp = ((1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))) + (1.0d0 + ((x * 0.5d0) - 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 tmp;
	if (y <= 0.00047) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + ((1.0 - Math.sqrt(x)) + (1.0 + ((y * (0.5 + (y * -0.125))) - Math.sqrt(y))));
	} else {
		tmp = ((1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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):
	tmp = 0
	if y <= 0.00047:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + ((1.0 - math.sqrt(x)) + (1.0 + ((y * (0.5 + (y * -0.125))) - math.sqrt(y))))
	else:
		tmp = ((1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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)
	tmp = 0.0
	if (y <= 0.00047)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 + Float64(Float64(y * Float64(0.5 + Float64(y * -0.125))) - sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))) + Float64(1.0 + Float64(Float64(x * 0.5) - 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)
	tmp = 0.0;
	if (y <= 0.00047)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((1.0 - sqrt(x)) + (1.0 + ((y * (0.5 + (y * -0.125))) - sqrt(y))));
	else
		tmp = ((1.0 / (sqrt(y) + sqrt((1.0 + y)))) + (1.0 + ((x * 0.5) - 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_] := If[LessEqual[y, 0.00047], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(y * N[(0.5 + N[(y * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $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 y < 4.69999999999999986e-4

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 46.9%

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

   6. Taylor expanded in t around inf 26.6%

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

   7. Taylor expanded in y around 0 26.6%

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

      1. associate--l+ 20.5%

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

   9. Simplified 26.6%

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

   if 4.69999999999999986e-4 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

       1. flip-- 86.3%

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

       2. div-inv 86.3%

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

       3. add-sqr-sqrt 51.7%

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

       4. add-sqr-sqrt 86.8%

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

       5. associate--l+ 91.2%

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

   11. Applied egg-rr 16.1%

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

       1. +-inverses 91.2%

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

       2. metadata-eval 91.2%

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

       3. *-lft-identity 91.2%

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

       4. +-commutative 91.2%

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

   13. Simplified 16.1%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00047:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(1 + \left(y \cdot \left(0.5 + y \cdot -0.125\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2800000000000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2800000000000.0)
   (+ (- (sqrt (+ z 1.0)) (sqrt z)) (- (- 2.0 (sqrt y)) (sqrt x)))
   (+
    (* 0.5 (sqrt (/ 1.0 t)))
    (+ 1.0 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2800000000000.0) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((2.0 - sqrt(y)) - sqrt(x));
	} else {
		tmp = (0.5 * sqrt((1.0 / t))) + (1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2800000000000.0d0) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + ((2.0d0 - sqrt(y)) - sqrt(x))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / t))) + (1.0d0 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2800000000000.0) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(y)) - Math.sqrt(x));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / t))) + (1.0 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 2800000000000.0:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + ((2.0 - math.sqrt(y)) - math.sqrt(x))
	else:
		tmp = (0.5 * math.sqrt((1.0 / t))) + (1.0 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2800000000000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) + Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2800000000000.0)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((2.0 - sqrt(y)) - sqrt(x));
	else
		tmp = (0.5 * sqrt((1.0 / t))) + (1.0 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 2800000000000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8e12

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. sub-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around 0 39.4%

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

   6. Taylor expanded in t around inf 24.2%

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

   7. Taylor expanded in y around 0 11.0%

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

      1. +-commutative 11.0%

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

      2. associate--r+ 11.0%

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

   9. Simplified 11.0%

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

   if 2.8e12 < z

   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. Step-by-step derivation

      1. associate-+l+ 83.2%

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

      2. sub-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. +-commutative 83.2%

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

      5. +-commutative 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in t around inf 40.3%

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

   6. Taylor expanded in x around 0 28.1%

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

      1. associate--l+ 28.1%

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

   8. Simplified 28.1%

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

   9. Taylor expanded in z around inf 28.1%

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

   10. Taylor expanded in x around 0 20.4%

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

       1. associate--l+ 26.4%

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

       2. +-commutative 26.4%

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

   12. Simplified 26.4%

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

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2800000000000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;z \leq 3000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(1 + \left(\sqrt{1 + y} - t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt y))))
   (if (<= z 3000000000000.0)
     (+ (- (sqrt (+ z 1.0)) (sqrt z)) (+ 2.0 (- (* y 0.5) t_1)))
     (+ (* 0.5 (sqrt (/ 1.0 t))) (+ 1.0 (- (sqrt (+ 1.0 y)) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(x) + sqrt(y);
	double tmp;
	if (z <= 3000000000000.0) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - t_1));
	} else {
		tmp = (0.5 * sqrt((1.0 / t))) + (1.0 + (sqrt((1.0 + y)) - t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(x) + sqrt(y)
    if (z <= 3000000000000.0d0) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + (2.0d0 + ((y * 0.5d0) - t_1))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / t))) + (1.0d0 + (sqrt((1.0d0 + y)) - t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt(y);
	double tmp;
	if (z <= 3000000000000.0) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (2.0 + ((y * 0.5) - t_1));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / t))) + (1.0 + (Math.sqrt((1.0 + y)) - t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(x) + math.sqrt(y)
	tmp = 0
	if z <= 3000000000000.0:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + (2.0 + ((y * 0.5) - t_1))
	else:
		tmp = (0.5 * math.sqrt((1.0 / t))) + (1.0 + (math.sqrt((1.0 + y)) - t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (z <= 3000000000000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(2.0 + Float64(Float64(y * 0.5) - t_1)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / t))) + Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(x) + sqrt(y);
	tmp = 0.0;
	if (z <= 3000000000000.0)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - t_1));
	else
		tmp = (0.5 * sqrt((1.0 / t))) + (1.0 + (sqrt((1.0 + y)) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3000000000000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 3e12

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. sub-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around 0 39.4%

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

   6. Taylor expanded in t around inf 24.2%

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

   7. Taylor expanded in y around 0 12.6%

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

      1. associate--l+ 12.6%

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

      2. *-commutative 12.6%

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

      3. +-commutative 12.6%

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

   9. Simplified 12.6%

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

   if 3e12 < z

   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. Step-by-step derivation

      1. associate-+l+ 83.2%

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

      2. sub-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. +-commutative 83.2%

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

      5. +-commutative 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in t around inf 40.3%

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

   6. Taylor expanded in x around 0 28.1%

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

      1. associate--l+ 28.1%

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

   8. Simplified 28.1%

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

   9. Taylor expanded in z around inf 28.1%

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

   10. Taylor expanded in x around 0 20.4%

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

       1. associate--l+ 26.4%

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

       2. +-commutative 26.4%

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

   12. Simplified 26.4%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3000000000000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1020000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{z}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1020000.0)
   (+
    (- (sqrt (+ z 1.0)) (sqrt z))
    (+ 2.0 (- (* y 0.5) (+ (sqrt x) (sqrt y)))))
   (+
    (* 0.5 (sqrt (/ 1.0 z)))
    (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- 1.0 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1020000.0) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = (0.5 * sqrt((1.0 / z))) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1020000.0d0) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + (2.0d0 + ((y * 0.5d0) - (sqrt(x) + sqrt(y))))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / z))) + ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1020000.0) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (2.0 + ((y * 0.5) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / z))) + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (1.0 - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1020000.0:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + (2.0 + ((y * 0.5) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = (0.5 * math.sqrt((1.0 / z))) + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (1.0 - math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1020000.0)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(2.0 + Float64(Float64(y * 0.5) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1020000.0)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + (2.0 + ((y * 0.5) - (sqrt(x) + sqrt(y))));
	else
		tmp = (0.5 * sqrt((1.0 / z))) + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1020000.0], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.02e6

   1. Initial program 97.6%

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

      1. associate-+l+ 97.6%

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

      2. sub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in t around inf 24.6%

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

   7. Taylor expanded in y around 0 12.0%

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

      1. associate--l+ 12.0%

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

      2. *-commutative 12.0%

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

      3. +-commutative 12.0%

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

   9. Simplified 12.0%

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

   if 1.02e6 < z

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in x around 0 49.8%

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

   6. Taylor expanded in t around inf 28.1%

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

   7. Taylor expanded in z around inf 28.4%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1020000:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(2 + \left(y \cdot 0.5 - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{z}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ t_2 := 1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;t\_1 + \left(t\_2 + \left(\left(1 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_2 + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))) (t_2 (+ 1.0 (- (* x 0.5) (sqrt x)))))
   (if (<= y 4.5e-21)
     (+ (- (sqrt (+ z 1.0)) (sqrt z)) (- (- 2.0 (sqrt y)) (sqrt x)))
     (if (<= y 1.3)
       (+
        t_1
        (+
         t_2
         (- (+ 1.0 (* y (+ 0.5 (* y (- (* y 0.0625) 0.125))))) (sqrt y))))
       (+ t_1 (+ t_2 (* 0.5 (sqrt (/ 1.0 y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double t_2 = 1.0 + ((x * 0.5) - sqrt(x));
	double tmp;
	if (y <= 4.5e-21) {
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((2.0 - sqrt(y)) - sqrt(x));
	} else if (y <= 1.3) {
		tmp = t_1 + (t_2 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - sqrt(y)));
	} else {
		tmp = t_1 + (t_2 + (0.5 * sqrt((1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / t))
    t_2 = 1.0d0 + ((x * 0.5d0) - sqrt(x))
    if (y <= 4.5d-21) then
        tmp = (sqrt((z + 1.0d0)) - sqrt(z)) + ((2.0d0 - sqrt(y)) - sqrt(x))
    else if (y <= 1.3d0) then
        tmp = t_1 + (t_2 + ((1.0d0 + (y * (0.5d0 + (y * ((y * 0.0625d0) - 0.125d0))))) - sqrt(y)))
    else
        tmp = t_1 + (t_2 + (0.5d0 * sqrt((1.0d0 / y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / t));
	double t_2 = 1.0 + ((x * 0.5) - Math.sqrt(x));
	double tmp;
	if (y <= 4.5e-21) {
		tmp = (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(y)) - Math.sqrt(x));
	} else if (y <= 1.3) {
		tmp = t_1 + (t_2 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - Math.sqrt(y)));
	} else {
		tmp = t_1 + (t_2 + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	t_2 = 1.0 + ((x * 0.5) - math.sqrt(x))
	tmp = 0
	if y <= 4.5e-21:
		tmp = (math.sqrt((z + 1.0)) - math.sqrt(z)) + ((2.0 - math.sqrt(y)) - math.sqrt(x))
	elif y <= 1.3:
		tmp = t_1 + (t_2 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - math.sqrt(y)))
	else:
		tmp = t_1 + (t_2 + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	t_2 = Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x)))
	tmp = 0.0
	if (y <= 4.5e-21)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(y)) - sqrt(x)));
	elseif (y <= 1.3)
		tmp = Float64(t_1 + Float64(t_2 + Float64(Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * 0.0625) - 0.125))))) - sqrt(y))));
	else
		tmp = Float64(t_1 + Float64(t_2 + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / t));
	t_2 = 1.0 + ((x * 0.5) - sqrt(x));
	tmp = 0.0;
	if (y <= 4.5e-21)
		tmp = (sqrt((z + 1.0)) - sqrt(z)) + ((2.0 - sqrt(y)) - sqrt(x));
	elseif (y <= 1.3)
		tmp = t_1 + (t_2 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - sqrt(y)));
	else
		tmp = t_1 + (t_2 + (0.5 * sqrt((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.5e-21], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3], N[(t$95$1 + N[(t$95$2 + N[(N[(1.0 + N[(y * N[(0.5 + N[(y * N[(N[(y * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$2 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.49999999999999968e-21

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 46.4%

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

   6. Taylor expanded in t around inf 26.7%

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

   7. Taylor expanded in y around 0 26.7%

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

      1. +-commutative 26.7%

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

      2. associate--r+ 26.7%

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

   9. Simplified 26.7%

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

   if 4.49999999999999968e-21 < y < 1.30000000000000004

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 11.7%

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

   6. Taylor expanded in x around 0 11.7%

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

      1. associate--l+ 11.7%

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

   8. Simplified 11.7%

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

   9. Taylor expanded in z around inf 11.7%

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

   10. Taylor expanded in y around 0 11.7%

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

   if 1.30000000000000004 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + \left(\left(1 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ t_2 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1.3:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(\left(1 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (- (* x 0.5) (sqrt x)))) (t_2 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.3)
     (+
      t_2
      (+ t_1 (- (+ 1.0 (* y (+ 0.5 (* y (- (* y 0.0625) 0.125))))) (sqrt y))))
     (+ t_2 (+ t_1 (* 0.5 (sqrt (/ 1.0 y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x * 0.5) - sqrt(x));
	double t_2 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.3) {
		tmp = t_2 + (t_1 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - sqrt(y)));
	} else {
		tmp = t_2 + (t_1 + (0.5 * sqrt((1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + ((x * 0.5d0) - sqrt(x))
    t_2 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.3d0) then
        tmp = t_2 + (t_1 + ((1.0d0 + (y * (0.5d0 + (y * ((y * 0.0625d0) - 0.125d0))))) - sqrt(y)))
    else
        tmp = t_2 + (t_1 + (0.5d0 * sqrt((1.0d0 / y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x * 0.5) - Math.sqrt(x));
	double t_2 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.3) {
		tmp = t_2 + (t_1 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - Math.sqrt(y)));
	} else {
		tmp = t_2 + (t_1 + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + ((x * 0.5) - math.sqrt(x))
	t_2 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.3:
		tmp = t_2 + (t_1 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - math.sqrt(y)))
	else:
		tmp = t_2 + (t_1 + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x)))
	t_2 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.3)
		tmp = Float64(t_2 + Float64(t_1 + Float64(Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * 0.0625) - 0.125))))) - sqrt(y))));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + ((x * 0.5) - sqrt(x));
	t_2 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.3)
		tmp = t_2 + (t_1 + ((1.0 + (y * (0.5 + (y * ((y * 0.0625) - 0.125))))) - sqrt(y)));
	else
		tmp = t_2 + (t_1 + (0.5 * sqrt((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.3], N[(t$95$2 + N[(t$95$1 + N[(N[(1.0 + N[(y * N[(0.5 + N[(y * N[(N[(y * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.30000000000000004

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

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

   if 1.30000000000000004 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + \left(\left(1 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1.3:\\ \;\;\;\;t\_1 + \left(\left(2 + \left(x \cdot 0.5 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.3)
     (+
      t_1
      (-
       (+ 2.0 (+ (* x 0.5) (* y (+ 0.5 (* y (- (* y 0.0625) 0.125))))))
       (+ (sqrt x) (sqrt y))))
     (+ t_1 (+ (+ 1.0 (- (* x 0.5) (sqrt x))) (* 0.5 (sqrt (/ 1.0 y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.3) {
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.3d0) then
        tmp = t_1 + ((2.0d0 + ((x * 0.5d0) + (y * (0.5d0 + (y * ((y * 0.0625d0) - 0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = t_1 + ((1.0d0 + ((x * 0.5d0) - sqrt(x))) + (0.5d0 * sqrt((1.0d0 / y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.3) {
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 + ((x * 0.5) - Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.3:
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = t_1 + ((1.0 + ((x * 0.5) - math.sqrt(x))) + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.3)
		tmp = Float64(t_1 + Float64(Float64(2.0 + Float64(Float64(x * 0.5) + Float64(y * Float64(0.5 + Float64(y * Float64(Float64(y * 0.0625) - 0.125)))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.3)
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * ((y * 0.0625) - 0.125)))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = t_1 + ((1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.3], N[(t$95$1 + N[(N[(2.0 + N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(0.5 + N[(y * N[(N[(y * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.30000000000000004

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

       \[\leadsto \color{blue}{\left(\left(2 + \left(0.5 \cdot x + y \cdot \left(0.5 + y \cdot \left(0.0625 \cdot y - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   if 1.30000000000000004 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(2 + \left(x \cdot 0.5 + y \cdot \left(0.5 + y \cdot \left(y \cdot 0.0625 - 0.125\right)\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ t_2 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1.25:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(1 + \left(y \cdot \left(0.5 + y \cdot -0.125\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (- (* x 0.5) (sqrt x)))) (t_2 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.25)
     (+ t_2 (+ t_1 (+ 1.0 (- (* y (+ 0.5 (* y -0.125))) (sqrt y)))))
     (+ t_2 (+ t_1 (* 0.5 (sqrt (/ 1.0 y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x * 0.5) - sqrt(x));
	double t_2 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.25) {
		tmp = t_2 + (t_1 + (1.0 + ((y * (0.5 + (y * -0.125))) - sqrt(y))));
	} else {
		tmp = t_2 + (t_1 + (0.5 * sqrt((1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + ((x * 0.5d0) - sqrt(x))
    t_2 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.25d0) then
        tmp = t_2 + (t_1 + (1.0d0 + ((y * (0.5d0 + (y * (-0.125d0)))) - sqrt(y))))
    else
        tmp = t_2 + (t_1 + (0.5d0 * sqrt((1.0d0 / y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x * 0.5) - Math.sqrt(x));
	double t_2 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.25) {
		tmp = t_2 + (t_1 + (1.0 + ((y * (0.5 + (y * -0.125))) - Math.sqrt(y))));
	} else {
		tmp = t_2 + (t_1 + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + ((x * 0.5) - math.sqrt(x))
	t_2 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.25:
		tmp = t_2 + (t_1 + (1.0 + ((y * (0.5 + (y * -0.125))) - math.sqrt(y))))
	else:
		tmp = t_2 + (t_1 + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x)))
	t_2 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.25)
		tmp = Float64(t_2 + Float64(t_1 + Float64(1.0 + Float64(Float64(y * Float64(0.5 + Float64(y * -0.125))) - sqrt(y)))));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + ((x * 0.5) - sqrt(x));
	t_2 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.25)
		tmp = t_2 + (t_1 + (1.0 + ((y * (0.5 + (y * -0.125))) - sqrt(y))));
	else
		tmp = t_2 + (t_1 + (0.5 * sqrt((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.25], N[(t$95$2 + N[(t$95$1 + N[(1.0 + N[(N[(y * N[(0.5 + N[(y * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

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

       1. associate--l+ 20.5%

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

   12. Simplified 20.5%

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

   if 1.25 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + \left(1 + \left(y \cdot \left(0.5 + y \cdot -0.125\right) - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1.2:\\ \;\;\;\;t\_1 + \left(\left(2 + \left(x \cdot 0.5 + y \cdot \left(0.5 + y \cdot -0.125\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.2)
     (+
      t_1
      (-
       (+ 2.0 (+ (* x 0.5) (* y (+ 0.5 (* y -0.125)))))
       (+ (sqrt x) (sqrt y))))
     (+ t_1 (+ (+ 1.0 (- (* x 0.5) (sqrt x))) (* 0.5 (sqrt (/ 1.0 y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.2) {
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * -0.125))))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.2d0) then
        tmp = t_1 + ((2.0d0 + ((x * 0.5d0) + (y * (0.5d0 + (y * (-0.125d0)))))) - (sqrt(x) + sqrt(y)))
    else
        tmp = t_1 + ((1.0d0 + ((x * 0.5d0) - sqrt(x))) + (0.5d0 * sqrt((1.0d0 / y))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.2) {
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * -0.125))))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = t_1 + ((1.0 + ((x * 0.5) - Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / y))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.2:
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * -0.125))))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = t_1 + ((1.0 + ((x * 0.5) - math.sqrt(x))) + (0.5 * math.sqrt((1.0 / y))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.2)
		tmp = Float64(t_1 + Float64(Float64(2.0 + Float64(Float64(x * 0.5) + Float64(y * Float64(0.5 + Float64(y * -0.125))))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.2)
		tmp = t_1 + ((2.0 + ((x * 0.5) + (y * (0.5 + (y * -0.125))))) - (sqrt(x) + sqrt(y)));
	else
		tmp = t_1 + ((1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.2], N[(t$95$1 + N[(N[(2.0 + N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(0.5 + N[(y * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999996

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

       \[\leadsto \color{blue}{\left(\left(2 + \left(0.5 \cdot x + y \cdot \left(0.5 + -0.125 \cdot y\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   if 1.19999999999999996 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(2 + \left(x \cdot 0.5 + y \cdot \left(0.5 + y \cdot -0.125\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

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

       1. associate--l+ 20.5%

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

       2. distribute-lft-out 20.5%

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

       3. +-commutative 20.5%

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

   12. Simplified 20.5%

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

   if 1 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. +-commutative 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t around inf 40.7%

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

   6. Taylor expanded in x around 0 25.9%

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

      1. associate--l+ 25.9%

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

   8. Simplified 25.9%

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

   9. Taylor expanded in z around inf 16.0%

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

   10. Taylor expanded in y around inf 15.6%

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

4. Final simplification 18.0%

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

Alternative 20: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;t\_1 + \left(2 + \left(0.5 \cdot \left(x + y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 2.4)
     (+ t_1 (+ 2.0 (- (* 0.5 (+ x y)) (+ (sqrt x) (sqrt y)))))
     (+ (+ 1.0 (- (* x 0.5) (sqrt x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 2.4) {
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = (1.0 + ((x * 0.5) - sqrt(x))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 2.4d0) then
        tmp = t_1 + (2.0d0 + ((0.5d0 * (x + y)) - (sqrt(x) + sqrt(y))))
    else
        tmp = (1.0d0 + ((x * 0.5d0) - sqrt(x))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 2.4) {
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = (1.0 + ((x * 0.5) - Math.sqrt(x))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 2.4:
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = (1.0 + ((x * 0.5) - math.sqrt(x))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(t_1 + Float64(2.0 + Float64(Float64(0.5 * Float64(x + y)) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 2.4)
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (sqrt(x) + sqrt(y))));
	else
		tmp = (1.0 + ((x * 0.5) - sqrt(x))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.4], N[(t$95$1 + N[(2.0 + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t around inf 51.4%

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

   6. Taylor expanded in x around 0 26.7%

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

      1. associate--l+ 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in z around inf 20.5%

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

   10. Taylor expanded in y around 0 20.5%

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

       1. associate--l+ 20.5%

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

       2. distribute-lft-out 20.5%

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

       3. +-commutative 20.5%

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

   12. Simplified 20.5%

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

   if 2.39999999999999991 < y

   1. Initial program 82.8%

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

      1. associate-+l+ 82.8%

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

      2. sub-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 40.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

   6. Taylor expanded in x around 0 25.9%

      \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   7. Step-by-step derivation

      1. associate--l+ 25.9%

         \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   9. Taylor expanded in z around inf 16.0%

      \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

   10. Taylor expanded in y around inf 15.5%

       \[\leadsto \color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
   11. Step-by-step derivation

       1. associate--l+ 15.5%

          \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   12. Simplified 15.5%

       \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(2 + \left(0.5 \cdot \left(x + y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1:\\ \;\;\;\;t\_1 + \left(2 + \left(0.5 \cdot \left(x + y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(1 + 0.5 \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.0)
     (+ t_1 (+ 2.0 (- (* 0.5 (+ x y)) (+ (sqrt x) (sqrt y)))))
     (+ t_1 (- (+ 1.0 (* 0.5 (+ x (sqrt (/ 1.0 y))))) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.0) {
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (sqrt(x) + sqrt(y))));
	} else {
		tmp = t_1 + ((1.0 + (0.5 * (x + sqrt((1.0 / 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 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.0d0) then
        tmp = t_1 + (2.0d0 + ((0.5d0 * (x + y)) - (sqrt(x) + sqrt(y))))
    else
        tmp = t_1 + ((1.0d0 + (0.5d0 * (x + sqrt((1.0d0 / 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 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.0) {
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (Math.sqrt(x) + Math.sqrt(y))));
	} else {
		tmp = t_1 + ((1.0 + (0.5 * (x + Math.sqrt((1.0 / y))))) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.0:
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (math.sqrt(x) + math.sqrt(y))))
	else:
		tmp = t_1 + ((1.0 + (0.5 * (x + math.sqrt((1.0 / y))))) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(t_1 + Float64(2.0 + Float64(Float64(0.5 * Float64(x + y)) - Float64(sqrt(x) + sqrt(y)))));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 + Float64(0.5 * Float64(x + sqrt(Float64(1.0 / 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 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.0)
		tmp = t_1 + (2.0 + ((0.5 * (x + y)) - (sqrt(x) + sqrt(y))));
	else
		tmp = t_1 + ((1.0 + (0.5 * (x + sqrt((1.0 / 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[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.0], N[(t$95$1 + N[(2.0 + N[(N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 + N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 51.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

   6. Taylor expanded in x around 0 26.7%

      \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   7. Step-by-step derivation

      1. associate--l+ 26.7%

         \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Simplified 26.7%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   9. Taylor expanded in z around inf 20.5%

      \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

   10. Taylor expanded in y around 0 20.5%

       \[\leadsto \color{blue}{\left(\left(2 + \left(0.5 \cdot x + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
   11. Step-by-step derivation

       1. associate--l+ 20.5%

          \[\leadsto \color{blue}{\left(2 + \left(\left(0.5 \cdot x + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

       2. distribute-lft-out 20.5%

          \[\leadsto \left(2 + \left(\color{blue}{0.5 \cdot \left(x + y\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]

       3. +-commutative 20.5%

          \[\leadsto \left(2 + \left(0.5 \cdot \left(x + y\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   12. Simplified 20.5%

       \[\leadsto \color{blue}{\left(2 + \left(0.5 \cdot \left(x + y\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   if 1 < y

   1. Initial program 82.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 82.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 40.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

   6. Taylor expanded in x around 0 25.9%

      \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   7. Step-by-step derivation

      1. associate--l+ 25.9%

         \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   9. Taylor expanded in z around inf 16.0%

      \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

   10. Taylor expanded in y around inf 15.6%

       \[\leadsto \color{blue}{\left(\left(1 + \left(0.5 \cdot x + 0.5 \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
   11. Step-by-step derivation

       1. distribute-lft-out 15.6%

          \[\leadsto \left(\left(1 + \color{blue}{0.5 \cdot \left(x + \sqrt{\frac{1}{y}}\right)}\right) - \sqrt{x}\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   12. Simplified 15.6%

       \[\leadsto \color{blue}{\left(\left(1 + 0.5 \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(2 + \left(0.5 \cdot \left(x + y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(1 + 0.5 \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;y \leq 1:\\ \;\;\;\;t\_1 + \left(\left(x \cdot 0.5 + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= y 1.0)
     (+ t_1 (- (+ (* x 0.5) 2.0) (+ (sqrt x) (sqrt y))))
     (+ (+ 1.0 (- (* x 0.5) (sqrt x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (y <= 1.0) {
		tmp = t_1 + (((x * 0.5) + 2.0) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (1.0 + ((x * 0.5) - sqrt(x))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / t))
    if (y <= 1.0d0) then
        tmp = t_1 + (((x * 0.5d0) + 2.0d0) - (sqrt(x) + sqrt(y)))
    else
        tmp = (1.0d0 + ((x * 0.5d0) - sqrt(x))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (y <= 1.0) {
		tmp = t_1 + (((x * 0.5) + 2.0) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (1.0 + ((x * 0.5) - Math.sqrt(x))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if y <= 1.0:
		tmp = t_1 + (((x * 0.5) + 2.0) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (1.0 + ((x * 0.5) - math.sqrt(x))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(t_1 + Float64(Float64(Float64(x * 0.5) + 2.0) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (y <= 1.0)
		tmp = t_1 + (((x * 0.5) + 2.0) - (sqrt(x) + sqrt(y)));
	else
		tmp = (1.0 + ((x * 0.5) - sqrt(x))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.0], N[(t$95$1 + N[(N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 51.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

   6. Taylor expanded in x around 0 26.7%

      \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   7. Step-by-step derivation

      1. associate--l+ 26.7%

         \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Simplified 26.7%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   9. Taylor expanded in z around inf 20.5%

      \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

   10. Taylor expanded in y around 0 20.5%

       \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   if 1 < y

   1. Initial program 82.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 82.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 82.8%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 40.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

   6. Taylor expanded in x around 0 25.9%

      \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
   7. Step-by-step derivation

      1. associate--l+ 25.9%

         \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

   9. Taylor expanded in z around inf 16.0%

      \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

   10. Taylor expanded in y around inf 15.5%

       \[\leadsto \color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
   11. Step-by-step derivation

       1. associate--l+ 15.5%

          \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

   12. Simplified 15.5%

       \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{t}} + \left(\left(x \cdot 0.5 + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 34.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(1 - \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ z 1.0)) (sqrt z)) (- 1.0 (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((z + 1.0)) - sqrt(z)) + (1.0 - sqrt(x));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((z + 1.0d0)) - sqrt(z)) + (1.0d0 - sqrt(x))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((z + 1.0)) - Math.sqrt(z)) + (1.0 - Math.sqrt(x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((z + 1.0)) - math.sqrt(z)) + (1.0 - math.sqrt(x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + 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 = (sqrt((z + 1.0)) - sqrt(z)) + (1.0 - sqrt(x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 89.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 45.5%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in t around inf 26.4%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

7. Taylor expanded in y around inf 17.5%

   \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

8. Final simplification 17.5%

   \[\leadsto \left(\sqrt{z + 1} - \sqrt{z}\right) + \left(1 - \sqrt{x}\right) \]
9. Add Preprocessing

Alternative 24: 34.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (+ 1.0 (- (* x 0.5) (sqrt x))) (* 0.5 (sqrt (/ 1.0 t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / t)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 + ((x * 0.5d0) - sqrt(x))) + (0.5d0 * sqrt((1.0d0 / t)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (1.0 + ((x * 0.5) - Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / t)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (1.0 + ((x * 0.5) - math.sqrt(x))) + (0.5 * math.sqrt((1.0 / t)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / t))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 + ((x * 0.5) - sqrt(x))) + (0.5 * sqrt((1.0 / t)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 89.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 46.0%

   \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}}\right) \]

6. Taylor expanded in x around 0 26.3%

   \[\leadsto \left(\color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]
7. Step-by-step derivation

   1. associate--l+ 26.3%

      \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

8. Simplified 26.3%

   \[\leadsto \left(\color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\right) \]

9. Taylor expanded in z around inf 18.2%

   \[\leadsto \left(\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

10. Taylor expanded in y around inf 12.2%

    \[\leadsto \color{blue}{\left(\left(1 + 0.5 \cdot x\right) - \sqrt{x}\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]
11. Step-by-step derivation

    1. associate--l+ 12.2%

       \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

12. Simplified 12.2%

    \[\leadsto \color{blue}{\left(1 + \left(0.5 \cdot x - \sqrt{x}\right)\right)} + 0.5 \cdot \sqrt{\frac{1}{t}} \]

13. Final simplification 12.2%

    \[\leadsto \left(1 + \left(x \cdot 0.5 - \sqrt{x}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}} \]
14. Add Preprocessing

Alternative 25: 9.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0048:\\ \;\;\;\;1 - \left(\sqrt{z} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{z}} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.0048)
   (- 1.0 (+ (sqrt z) (sqrt x)))
   (- (* 0.5 (sqrt (/ 1.0 z))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.0048) {
		tmp = 1.0 - (sqrt(z) + sqrt(x));
	} else {
		tmp = (0.5 * sqrt((1.0 / z))) - 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 <= 0.0048d0) then
        tmp = 1.0d0 - (sqrt(z) + sqrt(x))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / z))) - 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 <= 0.0048) {
		tmp = 1.0 - (Math.sqrt(z) + Math.sqrt(x));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / z))) - 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 <= 0.0048:
		tmp = 1.0 - (math.sqrt(z) + math.sqrt(x))
	else:
		tmp = (0.5 * math.sqrt((1.0 / z))) - 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 <= 0.0048)
		tmp = Float64(1.0 - Float64(sqrt(z) + sqrt(x)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) - 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 <= 0.0048)
		tmp = 1.0 - (sqrt(z) + sqrt(x));
	else
		tmp = (0.5 * sqrt((1.0 / z))) - 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, 0.0048], N[(1.0 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00479999999999999958

   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. Step-by-step derivation

      1. associate-+l+ 97.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 97.5%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 97.5%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 97.5%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 97.5%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 97.5%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 41.0%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 24.3%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in x around inf 8.3%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 8.3%

         \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 8.3%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around 0 8.3%

       \[\leadsto \color{blue}{1 - \left(\sqrt{x} + \sqrt{z}\right)} \]

   if 0.00479999999999999958 < z

   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. Step-by-step derivation

      1. associate-+l+ 83.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. sub-neg 83.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. sub-neg 83.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. +-commutative 83.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      5. +-commutative 83.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      6. +-commutative 83.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.4%

      \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Taylor expanded in t around inf 28.2%

      \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   7. Taylor expanded in x around inf 1.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 1.9%

         \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   9. Simplified 1.9%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   10. Taylor expanded in z around inf 3.1%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0048:\\ \;\;\;\;1 - \left(\sqrt{z} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{z}} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 9.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(1 + z \cdot 0.5\right) - \left(\sqrt{z} + \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (- (+ 1.0 (* z 0.5)) (+ (sqrt z) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (1.0 + (z * 0.5)) - (sqrt(z) + 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 + (z * 0.5d0)) - (sqrt(z) + 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 + (z * 0.5)) - (Math.sqrt(z) + Math.sqrt(x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (1.0 + (z * 0.5)) - (math.sqrt(z) + math.sqrt(x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(z * 0.5)) - Float64(sqrt(z) + sqrt(x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 + (z * 0.5)) - (sqrt(z) + sqrt(x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(1.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 89.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 45.5%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in t around inf 26.4%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 4.9%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

9. Simplified 4.9%

   \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

10. Taylor expanded in z around 0 6.3%

    \[\leadsto \color{blue}{\left(1 + 0.5 \cdot z\right) - \left(\sqrt{x} + \sqrt{z}\right)} \]

11. Final simplification 6.3%

    \[\leadsto \left(1 + z \cdot 0.5\right) - \left(\sqrt{z} + \sqrt{x}\right) \]
12. Add Preprocessing

Alternative 27: 6.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 - \left(\sqrt{z} + \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (+ (sqrt z) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - (sqrt(z) + 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(z) + 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(z) + Math.sqrt(x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - (math.sqrt(z) + math.sqrt(x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(sqrt(z) + 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(z) + sqrt(x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 89.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 45.5%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in t around inf 26.4%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 4.9%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

9. Simplified 4.9%

   \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

10. Taylor expanded in z around 0 4.7%

    \[\leadsto \color{blue}{1 - \left(\sqrt{x} + \sqrt{z}\right)} \]

11. Final simplification 4.7%

    \[\leadsto 1 - \left(\sqrt{z} + \sqrt{x}\right) \]
12. Add Preprocessing

Alternative 28: 2.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := (-N[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 89.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 89.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} + \left(-\sqrt{y}\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. sub-neg 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   5. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   6. +-commutative 89.9%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 45.5%

   \[\leadsto \left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Taylor expanded in t around inf 26.4%

   \[\leadsto \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 4.9%

      \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

9. Simplified 4.9%

   \[\leadsto \color{blue}{\left(-\sqrt{x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

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. Final simplification 1.6%

    \[\leadsto -\sqrt{x} \]
14. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024053 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))