Main:z from

Percentage Accurate: 92.1% → 97.8%

Time: 48.4s

Alternatives: 18

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.1% 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: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := t_3 - \sqrt{z}\\ t_5 := \sqrt{x + 1}\\ t_6 := \sqrt{x} + t_5\\ \mathbf{if}\;y \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;t_5 - \left(\sqrt{x} + \left(t_1 + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - t_3} - t_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, t_2 + \left(t_4 + t_1\right), t_6 \cdot \left({\left(t_2 + t_4\right)}^{2} - {t_1}^{2}\right)\right)}{t_6 \cdot \left(t_2 + \left(\left(\sqrt{y} + -1\right) - \left(\sqrt{z} - t_3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (- (sqrt t) (sqrt (+ t 1.0)))))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ x 1.0)))
        (t_6 (+ (sqrt x) t_5)))
   (if (<= y 6.5e+15)
     (-
      t_5
      (+ (sqrt x) (+ t_1 (- (/ (+ (- z z) 1.0) (- (- (sqrt z)) t_3)) t_2))))
     (/
      (fma
       1.0
       (+ t_2 (+ t_4 t_1))
       (* t_6 (- (pow (+ t_2 t_4) 2.0) (pow t_1 2.0))))
      (* t_6 (+ t_2 (- (+ (sqrt y) -1.0) (- (sqrt z) t_3))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + (sqrt(t) - sqrt((t + 1.0)));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((x + 1.0));
	double t_6 = sqrt(x) + t_5;
	double tmp;
	if (y <= 6.5e+15) {
		tmp = t_5 - (sqrt(x) + (t_1 + ((((z - z) + 1.0) / (-sqrt(z) - t_3)) - t_2)));
	} else {
		tmp = fma(1.0, (t_2 + (t_4 + t_1)), (t_6 * (pow((t_2 + t_4), 2.0) - pow(t_1, 2.0)))) / (t_6 * (t_2 + ((sqrt(y) + -1.0) - (sqrt(z) - t_3))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + Float64(sqrt(t) - sqrt(Float64(t + 1.0))))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(x + 1.0))
	t_6 = Float64(sqrt(x) + t_5)
	tmp = 0.0
	if (y <= 6.5e+15)
		tmp = Float64(t_5 - Float64(sqrt(x) + Float64(t_1 + Float64(Float64(Float64(Float64(z - z) + 1.0) / Float64(Float64(-sqrt(z)) - t_3)) - t_2))));
	else
		tmp = Float64(fma(1.0, Float64(t_2 + Float64(t_4 + t_1)), Float64(t_6 * Float64((Float64(t_2 + t_4) ^ 2.0) - (t_1 ^ 2.0)))) / Float64(t_6 * Float64(t_2 + Float64(Float64(sqrt(y) + -1.0) - Float64(sqrt(z) - t_3)))));
	end
	return 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[y], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[y, 6.5e+15], N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$1 + N[(N[(N[(N[(z - z), $MachinePrecision] + 1.0), $MachinePrecision] / N[((-N[Sqrt[z], $MachinePrecision]) - t$95$3), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(t$95$2 + N[(t$95$4 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * N[(N[Power[N[(t$95$2 + t$95$4), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * N[(t$95$2 + N[(N[(N[Sqrt[y], $MachinePrecision] + -1.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e15

   1. Initial program 96.5%

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

      1. associate-+l+ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. associate-+l+ 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-+l- 49.5%

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

   3. Simplified 49.5%

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

      1. flip-- 49.5%

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

      2. frac-2neg 49.5%

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

      3. add-sqr-sqrt 41.7%

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

      4. add-sqr-sqrt 49.7%

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

   5. Applied egg-rr 49.7%

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

      1. neg-sub0 49.7%

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

      2. +-inverses 49.7%

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

      3. associate--l+ 49.8%

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

      4. associate--r+ 49.8%

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

      5. +-inverses 49.8%

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

      6. metadata-eval 49.8%

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

      7. +-inverses 49.8%

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

      8. neg-sub0 49.8%

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

      9. +-inverses 49.8%

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

      10. metadata-eval 49.8%

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

      11. metadata-eval 49.8%

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

      12. neg-sub0 49.8%

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

      13. +-inverses 49.8%

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

      14. +-commutative 49.8%

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

      15. associate--r+ 49.8%

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

      16. +-inverses 49.8%

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

      17. neg-sub0 49.8%

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

   7. Simplified 49.8%

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

   if 6.5e15 < y

   1. Initial program 85.3%

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

      1. associate-+l+ 85.3%

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

      2. associate-+l+ 85.3%

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

      3. +-commutative 85.3%

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

      4. associate-+l+ 85.3%

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

      5. +-commutative 85.3%

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

      6. associate-+l- 52.3%

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

   3. Simplified 21.1%

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

   5. Applied egg-rr 17.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in t around 0 26.7%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right) + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - \sqrt{z + 1}} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right), \left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left({\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)}^{2}\right)\right)}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{y + 1} + \left(\left(\sqrt{y} + -1\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right)}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x} + \sqrt{x + 1}\\ t_4 := \sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\\ t_5 := t_1 - \sqrt{z}\\ \frac{\mathsf{fma}\left(1, t_2 + \left(\frac{\left(z + 1\right) - z}{t_1 + \sqrt{z}} + t_4\right), t_3 \cdot \left({\left(t_2 + t_5\right)}^{2} - {t_4}^{2}\right)\right)}{t_3 \cdot \left(t_2 + \left(t_5 + t_4\right)\right)} \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (+ (sqrt x) (sqrt (+ x 1.0))))
        (t_4 (+ (sqrt y) (- (sqrt t) (sqrt (+ t 1.0)))))
        (t_5 (- t_1 (sqrt z))))
   (/
    (fma
     1.0
     (+ t_2 (+ (/ (- (+ z 1.0) z) (+ t_1 (sqrt z))) t_4))
     (* t_3 (- (pow (+ t_2 t_5) 2.0) (pow t_4 2.0))))
    (* t_3 (+ t_2 (+ t_5 t_4))))))

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 t_2 = sqrt((y + 1.0));
	double t_3 = sqrt(x) + sqrt((x + 1.0));
	double t_4 = sqrt(y) + (sqrt(t) - sqrt((t + 1.0)));
	double t_5 = t_1 - sqrt(z);
	return fma(1.0, (t_2 + ((((z + 1.0) - z) / (t_1 + sqrt(z))) + t_4)), (t_3 * (pow((t_2 + t_5), 2.0) - pow(t_4, 2.0)))) / (t_3 * (t_2 + (t_5 + t_4)));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(sqrt(x) + sqrt(Float64(x + 1.0)))
	t_4 = Float64(sqrt(y) + Float64(sqrt(t) - sqrt(Float64(t + 1.0))))
	t_5 = Float64(t_1 - sqrt(z))
	return Float64(fma(1.0, Float64(t_2 + Float64(Float64(Float64(Float64(z + 1.0) - z) / Float64(t_1 + sqrt(z))) + t_4)), Float64(t_3 * Float64((Float64(t_2 + t_5) ^ 2.0) - (t_4 ^ 2.0)))) / Float64(t_3 * Float64(t_2 + Float64(t_5 + t_4))))
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]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 * N[(t$95$2 + N[(N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[Power[N[(t$95$2 + t$95$5), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(t$95$2 + N[(t$95$5 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

5. Applied egg-rr 50.2%

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

7. Simplified 56.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1, \sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left({\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)}^{2}\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)\right)}} \]
8. Step-by-step derivation

   1. flip-- 36.7%

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

   2. add-sqr-sqrt 30.2%

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

   3. add-sqr-sqrt 36.8%

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

9. Applied egg-rr 56.4%

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

10. Final simplification 56.4%

    \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{y + 1} + \left(\frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right), \left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left({\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)}^{2}\right)\right)}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right)\right)} \]

Alternative 3: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{x} + t_2\\ t_4 := \sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\\ t_5 := \sqrt{z + 1}\\ t_6 := t_5 - \sqrt{z}\\ \mathbf{if}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;t_2 - \left(\sqrt{x} + \left(t_4 + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - t_5} - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, t_1 + \left(t_6 + t_4\right), t_3 \cdot \left({\left(t_1 + t_6\right)}^{2} - {t_4}^{2}\right)\right)}{t_3 \cdot \left(t_1 + \left(\left(t_5 + \sqrt{y}\right) - \sqrt{z}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (+ (sqrt x) t_2))
        (t_4 (+ (sqrt y) (- (sqrt t) (sqrt (+ t 1.0)))))
        (t_5 (sqrt (+ z 1.0)))
        (t_6 (- t_5 (sqrt z))))
   (if (<= x 3.3e+15)
     (-
      t_2
      (+ (sqrt x) (+ t_4 (- (/ (+ (- z z) 1.0) (- (- (sqrt z)) t_5)) t_1))))
     (/
      (fma
       1.0
       (+ t_1 (+ t_6 t_4))
       (* t_3 (- (pow (+ t_1 t_6) 2.0) (pow t_4 2.0))))
      (* t_3 (+ t_1 (- (+ t_5 (sqrt y)) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt(x) + t_2;
	double t_4 = sqrt(y) + (sqrt(t) - sqrt((t + 1.0)));
	double t_5 = sqrt((z + 1.0));
	double t_6 = t_5 - sqrt(z);
	double tmp;
	if (x <= 3.3e+15) {
		tmp = t_2 - (sqrt(x) + (t_4 + ((((z - z) + 1.0) / (-sqrt(z) - t_5)) - t_1)));
	} else {
		tmp = fma(1.0, (t_1 + (t_6 + t_4)), (t_3 * (pow((t_1 + t_6), 2.0) - pow(t_4, 2.0)))) / (t_3 * (t_1 + ((t_5 + sqrt(y)) - sqrt(z))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(x) + t_2)
	t_4 = Float64(sqrt(y) + Float64(sqrt(t) - sqrt(Float64(t + 1.0))))
	t_5 = sqrt(Float64(z + 1.0))
	t_6 = Float64(t_5 - sqrt(z))
	tmp = 0.0
	if (x <= 3.3e+15)
		tmp = Float64(t_2 - Float64(sqrt(x) + Float64(t_4 + Float64(Float64(Float64(Float64(z - z) + 1.0) / Float64(Float64(-sqrt(z)) - t_5)) - t_1))));
	else
		tmp = Float64(fma(1.0, Float64(t_1 + Float64(t_6 + t_4)), Float64(t_3 * Float64((Float64(t_1 + t_6) ^ 2.0) - (t_4 ^ 2.0)))) / Float64(t_3 * Float64(t_1 + Float64(Float64(t_5 + sqrt(y)) - sqrt(z)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.3e+15], N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$4 + N[(N[(N[(N[(z - z), $MachinePrecision] + 1.0), $MachinePrecision] / N[((-N[Sqrt[z], $MachinePrecision]) - t$95$5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(t$95$1 + N[(t$95$6 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[Power[N[(t$95$1 + t$95$6), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(t$95$1 + N[(N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. associate-+l+ 97.3%

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

      5. +-commutative 97.3%

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

      6. associate-+l- 97.3%

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

   3. Simplified 68.0%

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

      1. flip-- 68.0%

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

      2. frac-2neg 68.0%

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

      3. add-sqr-sqrt 55.3%

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

      4. add-sqr-sqrt 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. neg-sub0 68.3%

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

      2. +-inverses 68.3%

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

      3. associate--l+ 68.4%

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

      4. associate--r+ 68.4%

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

      5. +-inverses 68.4%

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

      6. metadata-eval 68.4%

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

      7. +-inverses 68.4%

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

      8. neg-sub0 68.4%

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

      9. +-inverses 68.4%

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

      10. metadata-eval 68.4%

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

      11. metadata-eval 68.4%

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

      12. neg-sub0 68.4%

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

      13. +-inverses 68.4%

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

      14. +-commutative 68.4%

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

      15. associate--r+ 68.4%

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

      16. +-inverses 68.4%

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

      17. neg-sub0 68.4%

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

   7. Simplified 68.4%

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

   if 3.3e15 < x

   1. Initial program 86.0%

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

      1. associate-+l+ 86.0%

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

      2. associate-+l+ 86.0%

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

      3. +-commutative 86.0%

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

      4. associate-+l+ 86.0%

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

      5. +-commutative 86.0%

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

      6. associate-+l- 7.7%

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

   3. Simplified 7.7%

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

   5. Applied egg-rr 47.3%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in t around inf 34.6%

      \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left({\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)}^{2}\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \color{blue}{\left(\left(\sqrt{y} + \sqrt{1 + z}\right) - \sqrt{z}\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 34.6%

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

   10. Simplified 34.6%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right) + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - \sqrt{z + 1}} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right), \left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left({\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)}^{2}\right)\right)}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} + \sqrt{y}\right) - \sqrt{z}\right)\right)}\\ \end{array} \]

Alternative 4: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{x} + t_2\\ t_4 := \sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\\ t_5 := \sqrt{z + 1}\\ t_6 := t_5 - \sqrt{z}\\ \mathbf{if}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;t_2 - \left(\sqrt{x} + \left(t_4 + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - t_5} - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, t_1 + \left(\left(t_5 + \sqrt{y}\right) - \sqrt{z}\right), t_3 \cdot \left({\left(t_1 + t_6\right)}^{2} - {t_4}^{2}\right)\right)}{t_3 \cdot \left(t_1 + \left(t_6 + t_4\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (+ (sqrt x) t_2))
        (t_4 (+ (sqrt y) (- (sqrt t) (sqrt (+ t 1.0)))))
        (t_5 (sqrt (+ z 1.0)))
        (t_6 (- t_5 (sqrt z))))
   (if (<= x 3.3e+15)
     (-
      t_2
      (+ (sqrt x) (+ t_4 (- (/ (+ (- z z) 1.0) (- (- (sqrt z)) t_5)) t_1))))
     (/
      (fma
       1.0
       (+ t_1 (- (+ t_5 (sqrt y)) (sqrt z)))
       (* t_3 (- (pow (+ t_1 t_6) 2.0) (pow t_4 2.0))))
      (* t_3 (+ t_1 (+ t_6 t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt(x) + t_2;
	double t_4 = sqrt(y) + (sqrt(t) - sqrt((t + 1.0)));
	double t_5 = sqrt((z + 1.0));
	double t_6 = t_5 - sqrt(z);
	double tmp;
	if (x <= 3.3e+15) {
		tmp = t_2 - (sqrt(x) + (t_4 + ((((z - z) + 1.0) / (-sqrt(z) - t_5)) - t_1)));
	} else {
		tmp = fma(1.0, (t_1 + ((t_5 + sqrt(y)) - sqrt(z))), (t_3 * (pow((t_1 + t_6), 2.0) - pow(t_4, 2.0)))) / (t_3 * (t_1 + (t_6 + t_4)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(x) + t_2)
	t_4 = Float64(sqrt(y) + Float64(sqrt(t) - sqrt(Float64(t + 1.0))))
	t_5 = sqrt(Float64(z + 1.0))
	t_6 = Float64(t_5 - sqrt(z))
	tmp = 0.0
	if (x <= 3.3e+15)
		tmp = Float64(t_2 - Float64(sqrt(x) + Float64(t_4 + Float64(Float64(Float64(Float64(z - z) + 1.0) / Float64(Float64(-sqrt(z)) - t_5)) - t_1))));
	else
		tmp = Float64(fma(1.0, Float64(t_1 + Float64(Float64(t_5 + sqrt(y)) - sqrt(z))), Float64(t_3 * Float64((Float64(t_1 + t_6) ^ 2.0) - (t_4 ^ 2.0)))) / Float64(t_3 * Float64(t_1 + Float64(t_6 + t_4))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.3e+15], N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$4 + N[(N[(N[(N[(z - z), $MachinePrecision] + 1.0), $MachinePrecision] / N[((-N[Sqrt[z], $MachinePrecision]) - t$95$5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(t$95$1 + N[(N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[Power[N[(t$95$1 + t$95$6), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(t$95$1 + N[(t$95$6 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. associate-+l+ 97.3%

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

      5. +-commutative 97.3%

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

      6. associate-+l- 97.3%

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

   3. Simplified 68.0%

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

      1. flip-- 68.0%

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

      2. frac-2neg 68.0%

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

      3. add-sqr-sqrt 55.3%

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

      4. add-sqr-sqrt 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. neg-sub0 68.3%

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

      2. +-inverses 68.3%

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

      3. associate--l+ 68.4%

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

      4. associate--r+ 68.4%

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

      5. +-inverses 68.4%

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

      6. metadata-eval 68.4%

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

      7. +-inverses 68.4%

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

      8. neg-sub0 68.4%

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

      9. +-inverses 68.4%

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

      10. metadata-eval 68.4%

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

      11. metadata-eval 68.4%

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

      12. neg-sub0 68.4%

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

      13. +-inverses 68.4%

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

      14. +-commutative 68.4%

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

      15. associate--r+ 68.4%

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

      16. +-inverses 68.4%

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

      17. neg-sub0 68.4%

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

   7. Simplified 68.4%

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

   if 3.3e15 < x

   1. Initial program 86.0%

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

      1. associate-+l+ 86.0%

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

      2. associate-+l+ 86.0%

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

      3. +-commutative 86.0%

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

      4. associate-+l+ 86.0%

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

      5. +-commutative 86.0%

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

      6. associate-+l- 7.7%

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

   3. Simplified 7.7%

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

   5. Applied egg-rr 47.3%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in t around inf 50.1%

      \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{1 + y} + \color{blue}{\left(\left(\sqrt{y} + \sqrt{1 + z}\right) - \sqrt{z}\right)}, \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left({\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)}^{2}\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 34.6%

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

   10. Simplified 50.1%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right) + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - \sqrt{z + 1}} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{y + 1} + \left(\left(\sqrt{z + 1} + \sqrt{y}\right) - \sqrt{z}\right), \left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left({\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{2} - {\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)}^{2}\right)\right)}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{x} + t_2\\ t_4 := \sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\\ t_5 := \sqrt{z + 1}\\ t_6 := t_5 - \sqrt{z}\\ t_7 := t_1 + \left(t_6 + t_4\right)\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;t_2 - \left(\sqrt{x} + \left(t_4 + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - t_5} - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, t_7, t_3 \cdot \left({\left(t_1 + t_6\right)}^{2} - y\right)\right)}{t_3 \cdot t_7}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (+ (sqrt x) t_2))
        (t_4 (+ (sqrt y) (- (sqrt t) (sqrt (+ t 1.0)))))
        (t_5 (sqrt (+ z 1.0)))
        (t_6 (- t_5 (sqrt z)))
        (t_7 (+ t_1 (+ t_6 t_4))))
   (if (<= x 3.5e-5)
     (-
      t_2
      (+ (sqrt x) (+ t_4 (- (/ (+ (- z z) 1.0) (- (- (sqrt z)) t_5)) t_1))))
     (/ (fma 1.0 t_7 (* t_3 (- (pow (+ t_1 t_6) 2.0) y))) (* t_3 t_7)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt(x) + t_2;
	double t_4 = sqrt(y) + (sqrt(t) - sqrt((t + 1.0)));
	double t_5 = sqrt((z + 1.0));
	double t_6 = t_5 - sqrt(z);
	double t_7 = t_1 + (t_6 + t_4);
	double tmp;
	if (x <= 3.5e-5) {
		tmp = t_2 - (sqrt(x) + (t_4 + ((((z - z) + 1.0) / (-sqrt(z) - t_5)) - t_1)));
	} else {
		tmp = fma(1.0, t_7, (t_3 * (pow((t_1 + t_6), 2.0) - y))) / (t_3 * t_7);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(x) + t_2)
	t_4 = Float64(sqrt(y) + Float64(sqrt(t) - sqrt(Float64(t + 1.0))))
	t_5 = sqrt(Float64(z + 1.0))
	t_6 = Float64(t_5 - sqrt(z))
	t_7 = Float64(t_1 + Float64(t_6 + t_4))
	tmp = 0.0
	if (x <= 3.5e-5)
		tmp = Float64(t_2 - Float64(sqrt(x) + Float64(t_4 + Float64(Float64(Float64(Float64(z - z) + 1.0) / Float64(Float64(-sqrt(z)) - t_5)) - t_1))));
	else
		tmp = Float64(fma(1.0, t_7, Float64(t_3 * Float64((Float64(t_1 + t_6) ^ 2.0) - y))) / Float64(t_3 * t_7));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$1 + N[(t$95$6 + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5e-5], N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$4 + N[(N[(N[(N[(z - z), $MachinePrecision] + 1.0), $MachinePrecision] / N[((-N[Sqrt[z], $MachinePrecision]) - t$95$5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$7 + N[(t$95$3 * N[(N[Power[N[(t$95$1 + t$95$6), $MachinePrecision], 2.0], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$7), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999997e-5

   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. associate-+l+ 97.5%

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

      3. +-commutative 97.5%

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

      4. associate-+l+ 97.5%

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

      5. +-commutative 97.5%

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

      6. associate-+l- 97.5%

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

   3. Simplified 69.9%

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

      1. flip-- 69.9%

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

      2. frac-2neg 69.9%

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

      3. add-sqr-sqrt 56.7%

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

      4. add-sqr-sqrt 70.1%

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

   5. Applied egg-rr 70.1%

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

      1. neg-sub0 70.1%

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

      2. +-inverses 70.1%

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

      3. associate--l+ 70.2%

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

      4. associate--r+ 70.2%

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

      5. +-inverses 70.2%

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

      6. metadata-eval 70.2%

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

      7. +-inverses 70.2%

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

      8. neg-sub0 70.2%

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

      9. +-inverses 70.2%

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

      10. metadata-eval 70.2%

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

      11. metadata-eval 70.2%

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

      12. neg-sub0 70.2%

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

      13. +-inverses 70.2%

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

      14. +-commutative 70.2%

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

      15. associate--r+ 70.2%

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

      16. +-inverses 70.2%

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

      17. neg-sub0 70.2%

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

   7. Simplified 70.2%

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

   if 3.4999999999999997e-5 < x

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. associate-+l+ 86.1%

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

      5. +-commutative 86.1%

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

      6. associate-+l- 10.1%

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

   3. Simplified 7.9%

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

   5. Applied egg-rr 46.2%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in t around inf 19.5%

      \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \color{blue}{\left({\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \sqrt{z}\right)}^{2} - y\right)}\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r- 34.4%

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

   10. Simplified 34.4%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right) + \left(\frac{\left(z - z\right) + 1}{\left(-\sqrt{z}\right) - \sqrt{z + 1}} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right), \left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left({\left(\sqrt{y + 1} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{2} - y\right)\right)}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{y + 1} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t + 1}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{z} \leq 0.9999995:\\ \;\;\;\;t_2 - \left(\sqrt{x} + \left(\frac{-1}{t_1 + \sqrt{z}} + \left(\sqrt{y} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \left(\left(\left(t_1 + 1\right) - \sqrt{z}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ x 1.0))))
   (if (<= (- t_1 (sqrt z)) 0.9999995)
     (-
      t_2
      (+ (sqrt x) (+ (/ -1.0 (+ t_1 (sqrt z))) (- (sqrt y) (sqrt (+ y 1.0))))))
     (+
      (- t_2 (sqrt x))
      (+
       (/ 1.0 (+ (sqrt t) (sqrt (+ t 1.0))))
       (- (- (+ t_1 1.0) (sqrt z)) (sqrt y)))))))

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 t_2 = sqrt((x + 1.0));
	double tmp;
	if ((t_1 - sqrt(z)) <= 0.9999995) {
		tmp = t_2 - (sqrt(x) + ((-1.0 / (t_1 + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0)))));
	} else {
		tmp = (t_2 - sqrt(x)) + ((1.0 / (sqrt(t) + sqrt((t + 1.0)))) + (((t_1 + 1.0) - sqrt(z)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    if ((t_1 - sqrt(z)) <= 0.9999995d0) then
        tmp = t_2 - (sqrt(x) + (((-1.0d0) / (t_1 + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0d0)))))
    else
        tmp = (t_2 - sqrt(x)) + ((1.0d0 / (sqrt(t) + sqrt((t + 1.0d0)))) + (((t_1 + 1.0d0) - sqrt(z)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 0.9999995) {
		tmp = t_2 - (Math.sqrt(x) + ((-1.0 / (t_1 + Math.sqrt(z))) + (Math.sqrt(y) - Math.sqrt((y + 1.0)))));
	} else {
		tmp = (t_2 - Math.sqrt(x)) + ((1.0 / (Math.sqrt(t) + Math.sqrt((t + 1.0)))) + (((t_1 + 1.0) - Math.sqrt(z)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 0.9999995:
		tmp = t_2 - (math.sqrt(x) + ((-1.0 / (t_1 + math.sqrt(z))) + (math.sqrt(y) - math.sqrt((y + 1.0)))))
	else:
		tmp = (t_2 - math.sqrt(x)) + ((1.0 / (math.sqrt(t) + math.sqrt((t + 1.0)))) + (((t_1 + 1.0) - math.sqrt(z)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 0.9999995)
		tmp = Float64(t_2 - Float64(sqrt(x) + Float64(Float64(-1.0 / Float64(t_1 + sqrt(z))) + Float64(sqrt(y) - sqrt(Float64(y + 1.0))))));
	else
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(t + 1.0)))) + Float64(Float64(Float64(t_1 + 1.0) - sqrt(z)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 0.9999995)
		tmp = t_2 - (sqrt(x) + ((-1.0 / (t_1 + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0)))));
	else
		tmp = (t_2 - sqrt(x)) + ((1.0 / (sqrt(t) + sqrt((t + 1.0)))) + (((t_1 + 1.0) - sqrt(z)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.9999995], N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[(-1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 0.999999500000000041

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. +-commutative 86.7%

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

      4. associate-+l+ 86.7%

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

      5. +-commutative 86.7%

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

      6. associate-+l- 48.3%

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

   3. Simplified 38.4%

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

      1. flip-- 38.4%

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

      2. frac-2neg 38.4%

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

      3. add-sqr-sqrt 26.3%

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

      4. add-sqr-sqrt 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. neg-sub0 38.7%

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

      2. +-inverses 38.7%

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

      3. associate--l+ 38.7%

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

      4. associate--r+ 38.7%

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

      5. +-inverses 38.7%

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

      6. metadata-eval 38.7%

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

      7. +-inverses 38.7%

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

      8. neg-sub0 38.7%

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

      9. +-inverses 38.7%

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

      10. metadata-eval 38.7%

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

      11. metadata-eval 38.7%

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

      12. neg-sub0 38.7%

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

      13. +-inverses 38.7%

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

      14. +-commutative 38.7%

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

      15. associate--r+ 38.7%

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

      16. +-inverses 38.7%

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

      17. neg-sub0 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in t around inf 32.0%

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

      1. +-commutative 32.0%

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

      2. associate--l+ 32.8%

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

      3. +-commutative 32.8%

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

   10. Simplified 32.8%

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

   if 0.999999500000000041 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   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. associate-+l+ 96.9%

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

      3. +-commutative 96.9%

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

      4. +-commutative 96.9%

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

      5. associate-+r+ 96.9%

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

      6. sub-neg 96.9%

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

      7. sub-neg 96.9%

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

      8. +-commutative 96.9%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in y around 0 53.9%

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

      1. flip-- 53.9%

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

      2. div-inv 53.9%

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

      3. add-sqr-sqrt 42.3%

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

      4. add-sqr-sqrt 54.1%

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

      5. associate--l+ 54.2%

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

   6. Applied egg-rr 54.2%

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

      1. associate-*r/ 54.2%

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

      2. +-inverses 54.2%

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

      3. metadata-eval 54.2%

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

      4. metadata-eval 54.2%

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

   8. Simplified 54.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.9999995:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\frac{-1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y} - \sqrt{y + 1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \left(\left(\left(\sqrt{z + 1} + 1\right) - \sqrt{z}\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 7: 93.1% accurate, 1.0× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

   1. flip-- 36.7%

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

   2. frac-2neg 36.7%

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

   3. add-sqr-sqrt 30.2%

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

   4. add-sqr-sqrt 36.8%

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

5. Applied egg-rr 36.8%

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

   1. neg-sub0 36.8%

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

   2. +-inverses 36.8%

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

   3. associate--l+ 36.9%

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

   4. associate--r+ 36.9%

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

   5. +-inverses 36.9%

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

   6. metadata-eval 36.9%

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

   7. +-inverses 36.9%

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

   8. neg-sub0 36.9%

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

   9. +-inverses 36.9%

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

   10. metadata-eval 36.9%

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

   11. metadata-eval 36.9%

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

   12. neg-sub0 36.9%

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

   13. +-inverses 36.9%

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

   14. +-commutative 36.9%

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

   15. associate--r+ 36.9%

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

   16. +-inverses 36.9%

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

   17. neg-sub0 36.9%

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

7. Simplified 36.9%

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

8. Final simplification 36.9%

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

Alternative 8: 92.6% accurate, 1.0× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

   1. flip-- 36.7%

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

   2. add-sqr-sqrt 30.2%

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

   3. add-sqr-sqrt 36.8%

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

5. Applied egg-rr 36.8%

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

6. Final simplification 36.8%

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

Alternative 9: 92.5% accurate, 1.0× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

   1. flip-- 30.8%

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

   2. div-inv 30.8%

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

   3. add-sqr-sqrt 24.5%

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

   4. add-sqr-sqrt 30.9%

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

   5. associate--l+ 31.2%

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

5. Applied egg-rr 36.8%

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

   1. associate-*r/ 31.2%

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

   2. +-inverses 31.2%

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

   3. metadata-eval 31.2%

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

   4. metadata-eval 31.2%

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

7. Simplified 36.8%

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

8. Final simplification 36.8%

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

Alternative 10: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+17}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\frac{-1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y} - \sqrt{y + 1}\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= t 2.15e+17)
   (- (+ (sqrt (+ t 1.0)) 3.0) (+ (sqrt y) (sqrt t)))
   (-
    (sqrt (+ x 1.0))
    (+
     (sqrt x)
     (+
      (/ -1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))
      (- (sqrt y) (sqrt (+ y 1.0))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.15e+17) {
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	} else {
		tmp = sqrt((x + 1.0)) - (sqrt(x) + ((-1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.15d+17) then
        tmp = (sqrt((t + 1.0d0)) + 3.0d0) - (sqrt(y) + sqrt(t))
    else
        tmp = sqrt((x + 1.0d0)) - (sqrt(x) + (((-1.0d0) / (sqrt((z + 1.0d0)) + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0d0)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.15e+17) {
		tmp = (Math.sqrt((t + 1.0)) + 3.0) - (Math.sqrt(y) + Math.sqrt(t));
	} else {
		tmp = Math.sqrt((x + 1.0)) - (Math.sqrt(x) + ((-1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + (Math.sqrt(y) - Math.sqrt((y + 1.0)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 2.15e+17:
		tmp = (math.sqrt((t + 1.0)) + 3.0) - (math.sqrt(y) + math.sqrt(t))
	else:
		tmp = math.sqrt((x + 1.0)) - (math.sqrt(x) + ((-1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + (math.sqrt(y) - math.sqrt((y + 1.0)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.15e+17)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) + 3.0) - Float64(sqrt(y) + sqrt(t)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - Float64(sqrt(x) + Float64(Float64(-1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + Float64(sqrt(y) - sqrt(Float64(y + 1.0))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.15e+17)
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	else
		tmp = sqrt((x + 1.0)) - (sqrt(x) + ((-1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (sqrt(y) - sqrt((y + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 2.15e+17], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[(-1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.15e17

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-+r+ 95.7%

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

      6. sub-neg 95.7%

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

      7. sub-neg 95.7%

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

      8. +-commutative 95.7%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in y around 0 31.4%

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

   5. Taylor expanded in x around 0 17.2%

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

      1. associate-+r+ 17.2%

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

      2. +-commutative 17.2%

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

      3. +-commutative 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in z around 0 19.7%

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

   if 2.15e17 < t

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. associate-+l+ 86.6%

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

      3. +-commutative 86.6%

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

      4. associate-+l+ 86.6%

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

      5. +-commutative 86.6%

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

      6. associate-+l- 50.6%

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

   3. Simplified 45.3%

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

      1. flip-- 45.3%

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

      2. frac-2neg 45.3%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 45.6%

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

   5. Applied egg-rr 45.6%

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

      1. neg-sub0 45.6%

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

      2. +-inverses 45.6%

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

      3. associate--l+ 45.6%

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

      4. associate--r+ 45.6%

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

      5. +-inverses 45.6%

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

      6. metadata-eval 45.6%

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

      7. +-inverses 45.6%

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

      8. neg-sub0 45.6%

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

      9. +-inverses 45.6%

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

      10. metadata-eval 45.6%

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

      11. metadata-eval 45.6%

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

      12. neg-sub0 45.6%

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

      13. +-inverses 45.6%

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

      14. +-commutative 45.6%

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

      15. associate--r+ 45.6%

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

      16. +-inverses 45.6%

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

      17. neg-sub0 45.6%

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

   7. Simplified 45.6%

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

   8. Taylor expanded in t around inf 45.6%

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

      1. +-commutative 45.6%

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

      2. associate--l+ 51.3%

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

      3. +-commutative 51.3%

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

   10. Simplified 51.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+17}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{x} + \left(\frac{-1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y} - \sqrt{y + 1}\right)\right)\right)\\ \end{array} \]

Alternative 11: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 5e+18)
   (- (+ (sqrt (+ t 1.0)) 3.0) (+ (sqrt y) (sqrt t)))
   (+
    (sqrt (+ x 1.0))
    (-
     (- (sqrt (+ y 1.0)) (+ (sqrt y) (- (sqrt z) (sqrt (+ z 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 (t <= 5e+18) {
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	} else {
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 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 (t <= 5d+18) then
        tmp = (sqrt((t + 1.0d0)) + 3.0d0) - (sqrt(y) + sqrt(t))
    else
        tmp = sqrt((x + 1.0d0)) + ((sqrt((y + 1.0d0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 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 (t <= 5e+18) {
		tmp = (Math.sqrt((t + 1.0)) + 3.0) - (Math.sqrt(y) + Math.sqrt(t));
	} else {
		tmp = Math.sqrt((x + 1.0)) + ((Math.sqrt((y + 1.0)) - (Math.sqrt(y) + (Math.sqrt(z) - Math.sqrt((z + 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 t <= 5e+18:
		tmp = (math.sqrt((t + 1.0)) + 3.0) - (math.sqrt(y) + math.sqrt(t))
	else:
		tmp = math.sqrt((x + 1.0)) + ((math.sqrt((y + 1.0)) - (math.sqrt(y) + (math.sqrt(z) - math.sqrt((z + 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 (t <= 5e+18)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) + 3.0) - Float64(sqrt(y) + sqrt(t)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + Float64(sqrt(z) - sqrt(Float64(z + 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 (t <= 5e+18)
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	else
		tmp = sqrt((x + 1.0)) + ((sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 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[t, 5e+18], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5e18

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-+r+ 95.7%

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

      6. sub-neg 95.7%

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

      7. sub-neg 95.7%

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

      8. +-commutative 95.7%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in y around 0 31.4%

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

   5. Taylor expanded in x around 0 17.2%

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

      1. associate-+r+ 17.2%

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

      2. +-commutative 17.2%

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

      3. +-commutative 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in z around 0 19.7%

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

   if 5e18 < t

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. associate-+l+ 86.6%

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

      3. +-commutative 86.6%

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

      4. associate-+l+ 86.6%

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

      5. +-commutative 86.6%

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

      6. associate-+l- 50.6%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 31.8%

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

      1. associate--l+ 39.9%

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

      2. +-commutative 39.9%

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

      3. associate--l- 45.3%

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

   6. Simplified 45.3%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 12: 90.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 5e+18)
   (- (+ (sqrt (+ t 1.0)) 3.0) (+ (sqrt y) (sqrt t)))
   (+ (- (sqrt (+ y 1.0)) (+ (sqrt y) (- (sqrt z) (sqrt (+ z 1.0))))) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5e+18) {
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	} else {
		tmp = (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 1.0))))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5d+18) then
        tmp = (sqrt((t + 1.0d0)) + 3.0d0) - (sqrt(y) + sqrt(t))
    else
        tmp = (sqrt((y + 1.0d0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 1.0d0))))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5e+18) {
		tmp = (Math.sqrt((t + 1.0)) + 3.0) - (Math.sqrt(y) + Math.sqrt(t));
	} else {
		tmp = (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + (Math.sqrt(z) - Math.sqrt((z + 1.0))))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 5e+18:
		tmp = (math.sqrt((t + 1.0)) + 3.0) - (math.sqrt(y) + math.sqrt(t))
	else:
		tmp = (math.sqrt((y + 1.0)) - (math.sqrt(y) + (math.sqrt(z) - math.sqrt((z + 1.0))))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5e+18)
		tmp = Float64(Float64(sqrt(Float64(t + 1.0)) + 3.0) - Float64(sqrt(y) + sqrt(t)));
	else
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + Float64(sqrt(z) - sqrt(Float64(z + 1.0))))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 5e+18)
		tmp = (sqrt((t + 1.0)) + 3.0) - (sqrt(y) + sqrt(t));
	else
		tmp = (sqrt((y + 1.0)) - (sqrt(y) + (sqrt(z) - sqrt((z + 1.0))))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 5e+18], N[(N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5e18

   1. Initial program 95.7%

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

      1. associate-+l+ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-+r+ 95.7%

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

      6. sub-neg 95.7%

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

      7. sub-neg 95.7%

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

      8. +-commutative 95.7%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in y around 0 31.4%

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

   5. Taylor expanded in x around 0 17.2%

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

      1. associate-+r+ 17.2%

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

      2. +-commutative 17.2%

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

      3. +-commutative 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in z around 0 19.7%

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

   if 5e18 < t

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. associate-+l+ 86.6%

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

      3. +-commutative 86.6%

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

      4. associate-+l+ 86.6%

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

      5. +-commutative 86.6%

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

      6. associate-+l- 50.6%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 17.8%

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

      1. +-commutative 17.8%

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

      2. associate--l+ 31.6%

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

      3. +-commutative 31.6%

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

      4. +-commutative 31.6%

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

      5. +-commutative 31.6%

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

   6. Simplified 31.6%

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

      1. +-commutative 31.6%

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

      2. associate--r+ 31.8%

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

      3. associate-+l- 45.1%

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

      4. associate--r+ 39.3%

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

      5. associate-+r- 71.4%

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

      6. +-commutative 71.4%

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

      7. associate-+l- 39.3%

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

      8. associate--r+ 53.5%

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

      9. +-commutative 53.5%

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

   8. Applied egg-rr 53.5%

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

   9. Taylor expanded in x around 0 20.0%

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

       1. associate--l+ 56.3%

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

       2. associate--l+ 51.4%

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

       3. +-commutative 51.4%

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

       4. associate--r+ 56.6%

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

   11. Simplified 56.6%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{t + 1} + 3\right) - \left(\sqrt{y} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) + 1\\ \end{array} \]

Alternative 13: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.8e+14)
   (- (+ (sqrt (+ z 1.0)) 2.0) (+ (sqrt z) (sqrt y)))
   (+ (sqrt (+ x 1.0)) (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.8e+14) {
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2.8d+14) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - (sqrt(z) + sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) + (sqrt((y + 1.0d0)) - (sqrt(y) + sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.8e+14) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - (Math.sqrt(z) + Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 2.8e+14:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - (math.sqrt(z) + math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.8e+14)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.8e+14)
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(y));
	else
		tmp = sqrt((x + 1.0)) + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 2.8e+14], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8e14

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-+l+ 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. sub-neg 96.3%

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

      7. sub-neg 96.3%

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

      8. +-commutative 96.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in y around 0 52.1%

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

   5. Taylor expanded in x around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. +-commutative 18.0%

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

      3. +-commutative 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in t around inf 33.1%

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

   if 2.8e14 < z

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. +-commutative 86.7%

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

      4. associate-+l+ 86.7%

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

      5. +-commutative 86.7%

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

      6. associate-+l- 47.6%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around inf 3.8%

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

      1. +-commutative 3.8%

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

      2. associate--l+ 19.5%

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

      3. +-commutative 19.5%

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

      4. +-commutative 19.5%

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

      5. +-commutative 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in z around inf 20.4%

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

      1. associate--l+ 32.0%

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

      2. +-commutative 32.0%

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

      3. +-commutative 32.0%

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

   9. Simplified 32.0%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 14: 85.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.15e+14)
   (- (+ (sqrt (+ z 1.0)) 2.0) (+ (sqrt z) (sqrt y)))
   (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- (sqrt (+ x 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 <= 1.15e+14) {
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(y));
	} else {
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + (sqrt((x + 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 <= 1.15d+14) then
        tmp = (sqrt((z + 1.0d0)) + 2.0d0) - (sqrt(z) + sqrt(y))
    else
        tmp = (sqrt((y + 1.0d0)) - sqrt(y)) + (sqrt((x + 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 <= 1.15e+14) {
		tmp = (Math.sqrt((z + 1.0)) + 2.0) - (Math.sqrt(z) + Math.sqrt(y));
	} else {
		tmp = (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (Math.sqrt((x + 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 <= 1.15e+14:
		tmp = (math.sqrt((z + 1.0)) + 2.0) - (math.sqrt(z) + math.sqrt(y))
	else:
		tmp = (math.sqrt((y + 1.0)) - math.sqrt(y)) + (math.sqrt((x + 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 <= 1.15e+14)
		tmp = Float64(Float64(sqrt(Float64(z + 1.0)) + 2.0) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(sqrt(Float64(x + 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 <= 1.15e+14)
		tmp = (sqrt((z + 1.0)) + 2.0) - (sqrt(z) + sqrt(y));
	else
		tmp = (sqrt((y + 1.0)) - sqrt(y)) + (sqrt((x + 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, 1.15e+14], N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e14

   1. Initial program 96.3%

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

      1. associate-+l+ 96.3%

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

      2. associate-+l+ 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. sub-neg 96.3%

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

      7. sub-neg 96.3%

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

      8. +-commutative 96.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in y around 0 52.1%

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

   5. Taylor expanded in x around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. +-commutative 18.0%

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

      3. +-commutative 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in t around inf 33.1%

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

   if 1.15e14 < z

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. +-commutative 86.7%

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

      4. associate-+l+ 86.7%

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

      5. +-commutative 86.7%

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

      6. associate-+l- 47.6%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around inf 3.8%

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

      1. +-commutative 3.8%

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

      2. associate--l+ 19.5%

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

      3. +-commutative 19.5%

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

      4. +-commutative 19.5%

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

      5. +-commutative 19.5%

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

   6. Simplified 19.5%

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

      1. +-commutative 19.5%

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

      2. associate--r+ 19.2%

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

      3. associate-+l- 20.9%

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

      4. associate--r+ 14.9%

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

      5. associate-+r- 48.0%

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

      6. +-commutative 48.0%

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

      7. associate-+l- 14.9%

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

      8. associate--r+ 15.0%

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

      9. +-commutative 15.0%

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

   8. Applied egg-rr 15.0%

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

   9. Taylor expanded in z around inf 48.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{z + 1} + 2\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\\ \end{array} \]

Alternative 15: 82.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.6)
   (+ 2.0 (- (sqrt (+ z 1.0)) (+ (sqrt z) (sqrt y))))
   (- (sqrt (+ x 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 (y <= 1.6) {
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y)));
	} else {
		tmp = sqrt((x + 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 (y <= 1.6d0) then
        tmp = 2.0d0 + (sqrt((z + 1.0d0)) - (sqrt(z) + sqrt(y)))
    else
        tmp = sqrt((x + 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 (y <= 1.6) {
		tmp = 2.0 + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + Math.sqrt(y)));
	} else {
		tmp = Math.sqrt((x + 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 y <= 1.6:
		tmp = 2.0 + (math.sqrt((z + 1.0)) - (math.sqrt(z) + math.sqrt(y)))
	else:
		tmp = math.sqrt((x + 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 (y <= 1.6)
		tmp = Float64(2.0 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(sqrt(Float64(x + 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 (y <= 1.6)
		tmp = 2.0 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y)));
	else
		tmp = sqrt((x + 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[y, 1.6], N[(2.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6000000000000001

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+r+ 97.3%

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

      6. sub-neg 97.3%

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

      7. sub-neg 97.3%

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

      8. +-commutative 97.3%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in y around 0 56.8%

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

   5. Taylor expanded in x around 0 18.3%

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

      1. associate-+r+ 18.3%

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

      2. +-commutative 18.3%

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

      3. +-commutative 18.3%

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

   7. Simplified 18.3%

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

   8. Taylor expanded in t around inf 33.4%

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

      1. associate--l+ 59.3%

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

   10. Simplified 59.3%

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

   if 1.6000000000000001 < y

   1. Initial program 84.8%

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

      1. associate-+l+ 84.8%

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

      2. associate-+l+ 84.8%

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

      3. +-commutative 84.8%

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

      4. associate-+l+ 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+l- 51.0%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in t around inf 3.9%

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

      1. +-commutative 3.9%

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

      2. associate--l+ 20.7%

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

      3. +-commutative 20.7%

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

      4. +-commutative 20.7%

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

      5. +-commutative 20.7%

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

   6. Simplified 20.7%

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

   7. Taylor expanded in y around inf 26.2%

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

      1. +-commutative 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in z around inf 20.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 16: 61.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 2.8:\\ \;\;\;\;\left(t_1 + 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= y 2.8) (- (+ t_1 1.0) (sqrt x)) (- t_1 (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if (y <= 2.8) {
		tmp = (t_1 + 1.0) - sqrt(x);
	} else {
		tmp = t_1 - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if (y <= 2.8d0) then
        tmp = (t_1 + 1.0d0) - sqrt(x)
    else
        tmp = t_1 - 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((x + 1.0));
	double tmp;
	if (y <= 2.8) {
		tmp = (t_1 + 1.0) - Math.sqrt(x);
	} else {
		tmp = t_1 - 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((x + 1.0))
	tmp = 0
	if y <= 2.8:
		tmp = (t_1 + 1.0) - math.sqrt(x)
	else:
		tmp = t_1 - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 2.8)
		tmp = Float64(Float64(t_1 + 1.0) - sqrt(x));
	else
		tmp = Float64(t_1 - 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((x + 1.0));
	tmp = 0.0;
	if (y <= 2.8)
		tmp = (t_1 + 1.0) - sqrt(x);
	else
		tmp = t_1 - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.8], N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999998

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. associate-+l+ 97.3%

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

      5. +-commutative 97.3%

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

      6. associate-+l- 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in t around inf 17.8%

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

      1. +-commutative 17.8%

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

      2. associate--l+ 21.5%

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

      3. +-commutative 21.5%

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

      4. +-commutative 21.5%

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

      5. +-commutative 21.5%

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

   6. Simplified 21.5%

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

   7. Taylor expanded in y around inf 11.3%

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

      1. +-commutative 11.3%

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

   9. Simplified 11.3%

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

   10. Taylor expanded in z around 0 23.7%

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

   if 2.7999999999999998 < y

   1. Initial program 84.8%

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

      1. associate-+l+ 84.8%

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

      2. associate-+l+ 84.8%

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

      3. +-commutative 84.8%

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

      4. associate-+l+ 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+l- 51.0%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in t around inf 3.9%

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

      1. +-commutative 3.9%

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

      2. associate--l+ 20.7%

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

      3. +-commutative 20.7%

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

      4. +-commutative 20.7%

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

      5. +-commutative 20.7%

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

   6. Simplified 20.7%

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

   7. Taylor expanded in y around inf 26.2%

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

      1. +-commutative 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in z around inf 20.2%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8:\\ \;\;\;\;\left(\sqrt{x + 1} + 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 17: 35.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

4. Taylor expanded in t around inf 11.3%

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

   1. +-commutative 11.3%

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

   2. associate--l+ 21.1%

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

   3. +-commutative 21.1%

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

   4. +-commutative 21.1%

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

   5. +-commutative 21.1%

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

6. Simplified 21.1%

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

7. Taylor expanded in y around inf 18.3%

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

   1. +-commutative 18.3%

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

9. Simplified 18.3%

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

10. Taylor expanded in z around inf 14.7%

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

11. Final simplification 14.7%

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

Alternative 18: 34.3% accurate, 823.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. associate-+l+ 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-+l- 50.8%

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

3. Simplified 36.7%

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

4. Taylor expanded in t around inf 11.3%

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

   1. +-commutative 11.3%

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

   2. associate--l+ 21.1%

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

   3. +-commutative 21.1%

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

   4. +-commutative 21.1%

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

   5. +-commutative 21.1%

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

6. Simplified 21.1%

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

7. Taylor expanded in y around inf 18.3%

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

   1. +-commutative 18.3%

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

9. Simplified 18.3%

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

10. Taylor expanded in z around inf 14.7%

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

11. Taylor expanded in x around 0 37.1%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 37.1%

    \[\leadsto 1 \]

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

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