Main:z from

Percentage Accurate: 91.4% → 99.3%

Time: 29.7s

Alternatives: 19

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 0.2:\\ \;\;\;\;\left(t_3 - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(t_1 + \frac{1 + \left(x - x\right)}{\sqrt{x} + t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 + \left(t_1 - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{z} + t_3}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z))))
   (if (<= (- t_2 (sqrt x)) 0.2)
     (+
      (- t_3 (sqrt z))
      (expm1 (log1p (+ t_1 (/ (+ 1.0 (- x x)) (+ (sqrt x) t_2))))))
     (+
      (+ t_2 (- t_1 (sqrt x)))
      (+
       (- (sqrt (+ 1.0 t)) (sqrt t))
       (/ (+ 1.0 (- z z)) (+ (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 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.2) {
		tmp = (t_3 - sqrt(z)) + expm1(log1p((t_1 + ((1.0 + (x - x)) / (sqrt(x) + t_2)))));
	} else {
		tmp = (t_2 + (t_1 - sqrt(x))) + ((sqrt((1.0 + t)) - sqrt(t)) + ((1.0 + (z - z)) / (sqrt(z) + t_3)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.2) {
		tmp = (t_3 - Math.sqrt(z)) + Math.expm1(Math.log1p((t_1 + ((1.0 + (x - x)) / (Math.sqrt(x) + t_2)))));
	} else {
		tmp = (t_2 + (t_1 - Math.sqrt(x))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((1.0 + (z - z)) / (Math.sqrt(z) + t_3)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.2:
		tmp = (t_3 - math.sqrt(z)) + math.expm1(math.log1p((t_1 + ((1.0 + (x - x)) / (math.sqrt(x) + t_2)))))
	else:
		tmp = (t_2 + (t_1 - math.sqrt(x))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + ((1.0 + (z - z)) / (math.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(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.2)
		tmp = Float64(Float64(t_3 - sqrt(z)) + expm1(log1p(Float64(t_1 + Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_2))))));
	else
		tmp = Float64(Float64(t_2 + Float64(t_1 - sqrt(x))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 + Float64(z - z)) / 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[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[Log[1 + N[(t$95$1 + N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.20000000000000001

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. associate-+l- 47.6%

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

      3. +-commutative 47.6%

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

      4. sub-neg 47.6%

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

      5. sub-neg 47.6%

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

      6. +-commutative 47.6%

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

      7. +-commutative 47.6%

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

   3. Simplified 47.6%

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

      1. expm1-log1p-u 47.6%

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

      2. associate--r- 88.9%

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

      3. +-commutative 88.9%

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

   5. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. flip-- 88.9%

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

      2. add-sqr-sqrt 53.6%

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

      3. +-commutative 53.6%

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

      4. add-sqr-sqrt 89.9%

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

      5. +-commutative 89.9%

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

   7. Applied egg-rr 89.9%

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

      1. +-commutative 89.9%

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

      2. associate--l+ 92.1%

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

      3. +-commutative 92.1%

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

      4. +-commutative 92.1%

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

   9. Simplified 92.1%

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

       1. flip-- 92.1%

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

       2. add-sqr-sqrt 73.5%

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

       3. add-sqr-sqrt 92.6%

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

   11. Applied egg-rr 92.6%

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

       1. associate--l+ 94.5%

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

       2. +-inverses 94.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. metadata-eval 94.5%

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

   13. Simplified 94.5%

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

   14. Taylor expanded in t around inf 58.1%

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

   if 0.20000000000000001 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. associate-+l- 98.0%

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

      3. +-commutative 98.0%

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

      4. sub-neg 98.0%

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

      5. sub-neg 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

   3. Simplified 98.0%

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

      1. flip-- 97.4%

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

      2. add-sqr-sqrt 79.5%

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

      3. add-sqr-sqrt 97.5%

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

   5. Applied egg-rr 98.2%

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

      1. associate--l+ 97.7%

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

      2. +-inverses 97.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      3. metadata-eval 97.7%

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

   7. Simplified 98.4%

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

      1. flip-- 98.9%

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

      2. add-sqr-sqrt 75.1%

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

      3. +-commutative 75.1%

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

      4. add-sqr-sqrt 99.2%

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

      5. +-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. associate--l+ 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. associate-+r- 99.2%

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

       2. +-commutative 99.2%

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

       3. associate--l+ 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.2:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ t_4 := \left(\left(t_2 - \sqrt{y}\right) - \left(\sqrt{x} - t_1\right)\right) - \left(\sqrt{z} - t_3\right)\\ \mathbf{if}\;t_4 \leq 0.2:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \mathbf{elif}\;t_4 \leq 2.05:\\ \;\;\;\;t_1 + \left(\frac{1}{t_2 + \sqrt{y}} + \left(\frac{z - \left(z + -1\right)}{\sqrt{z} + t_3} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t_2\right) - \sqrt{y}\right) + \left(\left(t_3 - \sqrt{z}\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- (- (- t_2 (sqrt y)) (- (sqrt x) t_1)) (- (sqrt z) t_3))))
   (if (<= t_4 0.2)
     (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))
     (if (<= t_4 2.05)
       (+
        t_1
        (+
         (/ 1.0 (+ t_2 (sqrt y)))
         (- (/ (- z (+ z -1.0)) (+ (sqrt z) t_3)) (sqrt x))))
       (+
        (- (+ 1.0 t_2) (sqrt y))
        (+
         (- t_3 (sqrt z))
         (/ (- (+ 1.0 t) t) (+ (sqrt t) (sqrt (+ 1.0 t))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + z));
	double t_4 = ((t_2 - sqrt(y)) - (sqrt(x) - t_1)) - (sqrt(z) - t_3);
	double tmp;
	if (t_4 <= 0.2) {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	} else if (t_4 <= 2.05) {
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) + (((z - (z + -1.0)) / (sqrt(z) + t_3)) - sqrt(x)));
	} else {
		tmp = ((1.0 + t_2) - sqrt(y)) + ((t_3 - sqrt(z)) + (((1.0 + t) - t) / (sqrt(t) + sqrt((1.0 + t)))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = sqrt((1.0d0 + z))
    t_4 = ((t_2 - sqrt(y)) - (sqrt(x) - t_1)) - (sqrt(z) - t_3)
    if (t_4 <= 0.2d0) then
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + t_1)
    else if (t_4 <= 2.05d0) then
        tmp = t_1 + ((1.0d0 / (t_2 + sqrt(y))) + (((z - (z + (-1.0d0))) / (sqrt(z) + t_3)) - sqrt(x)))
    else
        tmp = ((1.0d0 + t_2) - sqrt(y)) + ((t_3 - sqrt(z)) + (((1.0d0 + t) - t) / (sqrt(t) + sqrt((1.0d0 + t)))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = ((t_2 - Math.sqrt(y)) - (Math.sqrt(x) - t_1)) - (Math.sqrt(z) - t_3);
	double tmp;
	if (t_4 <= 0.2) {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	} else if (t_4 <= 2.05) {
		tmp = t_1 + ((1.0 / (t_2 + Math.sqrt(y))) + (((z - (z + -1.0)) / (Math.sqrt(z) + t_3)) - Math.sqrt(x)));
	} else {
		tmp = ((1.0 + t_2) - Math.sqrt(y)) + ((t_3 - Math.sqrt(z)) + (((1.0 + t) - t) / (Math.sqrt(t) + Math.sqrt((1.0 + t)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = math.sqrt((1.0 + z))
	t_4 = ((t_2 - math.sqrt(y)) - (math.sqrt(x) - t_1)) - (math.sqrt(z) - t_3)
	tmp = 0
	if t_4 <= 0.2:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	elif t_4 <= 2.05:
		tmp = t_1 + ((1.0 / (t_2 + math.sqrt(y))) + (((z - (z + -1.0)) / (math.sqrt(z) + t_3)) - math.sqrt(x)))
	else:
		tmp = ((1.0 + t_2) - math.sqrt(y)) + ((t_3 - math.sqrt(z)) + (((1.0 + t) - t) / (math.sqrt(t) + math.sqrt((1.0 + t)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) - Float64(sqrt(x) - t_1)) - Float64(sqrt(z) - t_3))
	tmp = 0.0
	if (t_4 <= 0.2)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	elseif (t_4 <= 2.05)
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(Float64(Float64(z - Float64(z + -1.0)) / Float64(sqrt(z) + t_3)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 + t_2) - sqrt(y)) + Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(Float64(1.0 + t) - t) / Float64(sqrt(t) + sqrt(Float64(1.0 + t))))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = sqrt((1.0 + z));
	t_4 = ((t_2 - sqrt(y)) - (sqrt(x) - t_1)) - (sqrt(z) - t_3);
	tmp = 0.0;
	if (t_4 <= 0.2)
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	elseif (t_4 <= 2.05)
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) + (((z - (z + -1.0)) / (sqrt(z) + t_3)) - sqrt(x)));
	else
		tmp = ((1.0 + t_2) - sqrt(y)) + ((t_3 - sqrt(z)) + (((1.0 + t) - t) / (sqrt(t) + sqrt((1.0 + t)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.2], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.05], N[(t$95$1 + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 0.20000000000000001

   1. Initial program 56.3%

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

      1. associate-+l+ 56.3%

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

      2. +-commutative 56.3%

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

      3. associate-+r- 56.3%

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

      4. associate-+l- 16.9%

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

      5. +-commutative 16.9%

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

      6. associate--l+ 16.9%

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

      7. +-commutative 16.9%

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

   3. Simplified 5.3%

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

   4. Taylor expanded in t around inf 5.7%

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

      1. +-commutative 5.7%

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

      2. +-commutative 5.7%

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

      3. associate--l+ 8.4%

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

   6. Simplified 8.4%

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

   7. Taylor expanded in z around inf 5.3%

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

      1. +-commutative 5.3%

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

   9. Simplified 5.3%

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

   10. Taylor expanded in y around inf 8.4%

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

       1. flip-- 56.2%

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

       2. add-sqr-sqrt 42.2%

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

       3. +-commutative 42.2%

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

       4. add-sqr-sqrt 58.5%

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

       5. +-commutative 58.5%

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

   12. Applied egg-rr 9.5%

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

       1. +-commutative 58.5%

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

       2. associate--l+ 67.9%

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

       3. +-commutative 67.9%

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

       4. +-commutative 67.9%

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

   14. Simplified 19.8%

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

   if 0.20000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) < 2.0499999999999998

   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. +-commutative 97.3%

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

      3. associate-+r- 68.4%

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

      4. associate-+l- 50.2%

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

      5. +-commutative 50.2%

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

      6. associate--l+ 50.2%

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

      7. +-commutative 50.2%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in t around inf 30.3%

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

      1. +-commutative 30.3%

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

      2. +-commutative 30.3%

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

      3. associate--l+ 31.7%

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

   6. Simplified 31.7%

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

      1. flip-- 31.9%

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

      2. add-sqr-sqrt 23.1%

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

      3. add-sqr-sqrt 31.9%

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

      4. +-commutative 31.9%

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

      5. +-commutative 31.9%

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

   8. Applied egg-rr 31.9%

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

      1. associate--r+ 31.9%

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

      2. +-commutative 31.9%

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

   10. Simplified 31.9%

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

       1. flip-- 97.7%

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

       2. add-sqr-sqrt 77.5%

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

       3. add-sqr-sqrt 98.1%

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

   12. Applied egg-rr 32.0%

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

       1. associate--l+ 98.3%

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

       2. +-inverses 98.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. metadata-eval 98.3%

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

   14. Simplified 32.2%

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

   if 2.0499999999999998 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)))

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 92.4%

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

      1. flip-- 92.4%

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

      2. add-sqr-sqrt 68.0%

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

      3. +-commutative 68.0%

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

      4. add-sqr-sqrt 92.5%

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

      5. +-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)\right) - \left(\sqrt{z} - \sqrt{1 + z}\right) \leq 0.2:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{elif}\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)\right) - \left(\sqrt{z} - \sqrt{1 + z}\right) \leq 2.05:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{z - \left(z + -1\right)}{\sqrt{z} + \sqrt{1 + z}} - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;\left(t_3 - \sqrt{y}\right) - \left(\sqrt{x} - t_1\right) \leq 1.999985:\\ \;\;\;\;\left(t_2 - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{t_3 + \sqrt{y}} + \frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \left(\sqrt{x} + \left(\sqrt{y} - t_3\right)\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ 1.0 y))))
   (if (<= (- (- t_3 (sqrt y)) (- (sqrt x) t_1)) 1.999985)
     (+
      (- t_2 (sqrt z))
      (expm1
       (log1p
        (+ (/ 1.0 (+ t_3 (sqrt y))) (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))
     (+
      (- t_1 (+ (sqrt x) (- (sqrt y) t_3)))
      (+
       (- (sqrt (+ 1.0 t)) (sqrt t))
       (/ (- (+ 1.0 z) z) (+ (sqrt z) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (((t_3 - sqrt(y)) - (sqrt(x) - t_1)) <= 1.999985) {
		tmp = (t_2 - sqrt(z)) + expm1(log1p(((1.0 / (t_3 + sqrt(y))) + ((1.0 + (x - x)) / (sqrt(x) + t_1)))));
	} else {
		tmp = (t_1 - (sqrt(x) + (sqrt(y) - t_3))) + ((sqrt((1.0 + t)) - sqrt(t)) + (((1.0 + z) - z) / (sqrt(z) + t_2)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (((t_3 - Math.sqrt(y)) - (Math.sqrt(x) - t_1)) <= 1.999985) {
		tmp = (t_2 - Math.sqrt(z)) + Math.expm1(Math.log1p(((1.0 / (t_3 + Math.sqrt(y))) + ((1.0 + (x - x)) / (Math.sqrt(x) + t_1)))));
	} else {
		tmp = (t_1 - (Math.sqrt(x) + (Math.sqrt(y) - t_3))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (((1.0 + z) - z) / (Math.sqrt(z) + t_2)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if ((t_3 - math.sqrt(y)) - (math.sqrt(x) - t_1)) <= 1.999985:
		tmp = (t_2 - math.sqrt(z)) + math.expm1(math.log1p(((1.0 / (t_3 + math.sqrt(y))) + ((1.0 + (x - x)) / (math.sqrt(x) + t_1)))))
	else:
		tmp = (t_1 - (math.sqrt(x) + (math.sqrt(y) - t_3))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (((1.0 + z) - z) / (math.sqrt(z) + t_2)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(Float64(t_3 - sqrt(y)) - Float64(sqrt(x) - t_1)) <= 1.999985)
		tmp = Float64(Float64(t_2 - sqrt(z)) + expm1(log1p(Float64(Float64(1.0 / Float64(t_3 + sqrt(y))) + Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1))))));
	else
		tmp = Float64(Float64(t_1 - Float64(sqrt(x) + Float64(sqrt(y) - t_3))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_2))));
	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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], 1.999985], N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 1.9999849999999999

   1. Initial program 91.6%

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

      1. associate-+l+ 91.6%

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

      2. associate-+l- 63.7%

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

      3. +-commutative 63.7%

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

      4. sub-neg 63.7%

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

      5. sub-neg 63.7%

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

      6. +-commutative 63.7%

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

      7. +-commutative 63.7%

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

   3. Simplified 63.7%

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

      1. expm1-log1p-u 63.7%

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

      2. associate--r- 91.6%

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

      3. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. flip-- 91.6%

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

      2. add-sqr-sqrt 67.8%

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

      3. +-commutative 67.8%

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

      4. add-sqr-sqrt 92.3%

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

      5. +-commutative 92.3%

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

   7. Applied egg-rr 92.3%

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

      1. +-commutative 92.3%

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

      2. associate--l+ 93.8%

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

      3. +-commutative 93.8%

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

      4. +-commutative 93.8%

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

   9. Simplified 93.8%

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

       1. flip-- 93.8%

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

       2. add-sqr-sqrt 69.7%

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

       3. add-sqr-sqrt 94.2%

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

   11. Applied egg-rr 94.2%

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

       1. associate--l+ 95.7%

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

       2. +-inverses 95.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. metadata-eval 95.7%

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

   13. Simplified 95.7%

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

   14. Taylor expanded in t around inf 61.0%

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

   if 1.9999849999999999 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)))

   1. Initial program 98.8%

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

      1. associate-+l+ 98.8%

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

      2. associate-+l- 98.8%

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

      3. +-commutative 98.8%

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

      4. sub-neg 98.8%

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

      5. sub-neg 98.8%

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

      6. +-commutative 98.8%

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

      7. +-commutative 98.8%

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

   3. Simplified 98.8%

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

      1. flip-- 99.2%

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

      2. add-sqr-sqrt 78.9%

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

      3. +-commutative 78.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. +-commutative 99.9%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{1 + x}\right) \leq 1.999985:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-+l+ 93.4%

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

   2. associate-+l- 72.2%

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

   3. +-commutative 72.2%

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

   4. sub-neg 72.2%

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

   5. sub-neg 72.2%

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

   6. +-commutative 72.2%

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

   7. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. expm1-log1p-u 71.9%

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

   2. associate--r- 93.0%

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

   3. +-commutative 93.0%

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

5. Applied egg-rr 93.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
6. Step-by-step derivation

   1. flip-- 93.0%

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

   2. add-sqr-sqrt 75.0%

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

   3. +-commutative 75.0%

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

   4. add-sqr-sqrt 93.5%

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

   5. +-commutative 93.5%

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

7. Applied egg-rr 93.5%

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

   1. +-commutative 93.5%

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

   2. associate--l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. +-commutative 94.7%

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

9. Simplified 94.7%

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

    1. flip-- 94.7%

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

    2. add-sqr-sqrt 76.4%

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

    3. add-sqr-sqrt 95.0%

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

11. Applied egg-rr 95.0%

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

    1. associate--l+ 96.1%

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

    2. +-inverses 96.1%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

    3. metadata-eval 96.1%

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

13. Simplified 96.1%

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

    1. expm1-log1p-u 96.4%

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

    2. frac-add 96.4%

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

    3. +-inverses 96.4%

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

    4. metadata-eval 96.4%

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

    5. *-un-lft-identity 96.4%

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

15. Applied egg-rr 96.4%

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

    1. associate-/r* 96.4%

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

    2. *-rgt-identity 96.4%

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

    3. associate-+l+ 96.4%

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

    4. +-commutative 96.4%

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

    5. +-commutative 96.4%

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

17. Simplified 96.4%

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

18. Final simplification 96.4%

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

Alternative 5: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-+l+ 93.4%

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

   2. associate-+l- 72.2%

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

   3. +-commutative 72.2%

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

   4. sub-neg 72.2%

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

   5. sub-neg 72.2%

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

   6. +-commutative 72.2%

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

   7. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. expm1-log1p-u 71.9%

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

   2. associate--r- 93.0%

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

   3. +-commutative 93.0%

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

5. Applied egg-rr 93.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
6. Step-by-step derivation

   1. flip-- 93.0%

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

   2. add-sqr-sqrt 75.0%

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

   3. +-commutative 75.0%

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

   4. add-sqr-sqrt 93.5%

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

   5. +-commutative 93.5%

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

7. Applied egg-rr 93.5%

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

   1. +-commutative 93.5%

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

   2. associate--l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. +-commutative 94.7%

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

9. Simplified 94.7%

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

    1. flip-- 94.7%

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

    2. add-sqr-sqrt 76.4%

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

    3. add-sqr-sqrt 95.0%

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

11. Applied egg-rr 95.0%

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

    1. associate--l+ 96.1%

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

    2. +-inverses 96.1%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

    3. metadata-eval 96.1%

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

13. Simplified 96.1%

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

    1. expm1-log1p-u 96.4%

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

    2. frac-add 96.4%

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

    3. +-inverses 96.4%

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

    4. metadata-eval 96.4%

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

    5. *-un-lft-identity 96.4%

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

15. Applied egg-rr 96.4%

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

    1. *-rgt-identity 96.4%

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

    2. associate-+l+ 96.4%

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

    3. +-commutative 96.4%

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

    4. *-commutative 96.4%

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

    5. +-commutative 96.4%

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

17. Simplified 96.4%

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

18. Final simplification 96.4%

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

Alternative 6: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;t_1 - \sqrt{y} \leq 0.9999995:\\ \;\;\;\;\left(t_2 - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{t_1 + \sqrt{y}} + \frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t_2}\right) + \left(\left(1 + t_1\right) - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= (- t_1 (sqrt y)) 0.9999995)
     (+
      (- t_2 (sqrt z))
      (expm1
       (log1p
        (+
         (/ 1.0 (+ t_1 (sqrt y)))
         (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x))))))))
     (+
      (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ (- (+ 1.0 z) z) (+ (sqrt z) t_2)))
      (- (+ 1.0 t_1) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if ((t_1 - sqrt(y)) <= 0.9999995) {
		tmp = (t_2 - sqrt(z)) + expm1(log1p(((1.0 / (t_1 + sqrt(y))) + ((1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)))))));
	} else {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) + (((1.0 + z) - z) / (sqrt(z) + t_2))) + ((1.0 + t_1) - sqrt(y));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_1 - Math.sqrt(y)) <= 0.9999995) {
		tmp = (t_2 - Math.sqrt(z)) + Math.expm1(Math.log1p(((1.0 / (t_1 + Math.sqrt(y))) + ((1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)))))));
	} else {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (((1.0 + z) - z) / (Math.sqrt(z) + t_2))) + ((1.0 + t_1) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_1 - math.sqrt(y)) <= 0.9999995:
		tmp = (t_2 - math.sqrt(z)) + math.expm1(math.log1p(((1.0 / (t_1 + math.sqrt(y))) + ((1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))))))
	else:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (((1.0 + z) - z) / (math.sqrt(z) + t_2))) + ((1.0 + t_1) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_1 - sqrt(y)) <= 0.9999995)
		tmp = Float64(Float64(t_2 - sqrt(z)) + expm1(log1p(Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_2))) + Float64(Float64(1.0 + t_1) - sqrt(y)));
	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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.9999995], N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. associate-+l- 85.6%

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

      3. +-commutative 85.6%

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

      4. sub-neg 85.6%

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

      5. sub-neg 85.6%

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

      6. +-commutative 85.6%

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

      7. +-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. expm1-log1p-u 85.6%

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

      2. associate--r- 87.8%

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

      3. +-commutative 87.8%

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

   5. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   6. Step-by-step derivation

      1. flip-- 87.8%

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

      2. add-sqr-sqrt 73.6%

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

      3. +-commutative 73.6%

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

      4. add-sqr-sqrt 88.3%

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

      5. +-commutative 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. +-commutative 88.3%

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

      2. associate--l+ 90.4%

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

      3. +-commutative 90.4%

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

      4. +-commutative 90.4%

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

   9. Simplified 90.4%

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

       1. flip-- 90.4%

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

       2. add-sqr-sqrt 51.5%

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

       3. add-sqr-sqrt 91.1%

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

   11. Applied egg-rr 91.1%

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

       1. associate--l+ 93.4%

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

       2. +-inverses 93.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. metadata-eval 93.4%

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

   13. Simplified 93.4%

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

   14. Taylor expanded in t around inf 54.2%

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

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

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 60.4%

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

      3. +-commutative 60.4%

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

      4. sub-neg 60.4%

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

      5. sub-neg 60.4%

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

      6. +-commutative 60.4%

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

      7. +-commutative 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in x around 0 55.5%

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

      1. flip-- 60.5%

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

      2. add-sqr-sqrt 49.1%

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

      3. +-commutative 49.1%

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

      4. add-sqr-sqrt 60.9%

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

      5. +-commutative 60.9%

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

   6. Applied egg-rr 55.9%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.9999995:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \end{array} \]

Alternative 7: 97.6% accurate, 0.8× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-+l+ 93.4%

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

   2. associate-+l- 72.2%

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

   3. +-commutative 72.2%

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

   4. sub-neg 72.2%

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

   5. sub-neg 72.2%

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

   6. +-commutative 72.2%

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

   7. +-commutative 72.2%

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

3. Simplified 72.2%

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

   1. expm1-log1p-u 71.9%

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

   2. associate--r- 93.0%

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

   3. +-commutative 93.0%

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

5. Applied egg-rr 93.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
6. Step-by-step derivation

   1. flip-- 93.0%

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

   2. add-sqr-sqrt 75.0%

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

   3. +-commutative 75.0%

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

   4. add-sqr-sqrt 93.5%

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

   5. +-commutative 93.5%

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

7. Applied egg-rr 93.5%

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

   1. +-commutative 93.5%

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

   2. associate--l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. +-commutative 94.7%

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

9. Simplified 94.7%

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

    1. flip-- 94.7%

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

    2. add-sqr-sqrt 76.4%

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

    3. add-sqr-sqrt 95.0%

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

11. Applied egg-rr 95.0%

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

    1. associate--l+ 96.1%

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

    2. +-inverses 96.1%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

    3. metadata-eval 96.1%

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

13. Simplified 96.1%

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

    1. expm1-log1p-u 96.4%

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

    2. add-exp-log 96.3%

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

    3. +-commutative 96.3%

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

    4. +-inverses 96.3%

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

    5. metadata-eval 96.3%

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

15. Applied egg-rr 96.3%

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

16. Final simplification 96.3%

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

Alternative 8: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(t_2 - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;t_1 - \left(\left(\sqrt{x} + \frac{-1}{\sqrt{z} + t_2}\right) + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= y 4.5e-27)
     (+ (+ (- t_2 (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t))) 2.0)
     (if (<= y 3.3e+28)
       (-
        t_1
        (+
         (+ (sqrt x) (/ -1.0 (+ (sqrt z) t_2)))
         (- (sqrt y) (sqrt (+ 1.0 y)))))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if (y <= 4.5e-27) {
		tmp = ((t_2 - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	} else if (y <= 3.3e+28) {
		tmp = t_1 - ((sqrt(x) + (-1.0 / (sqrt(z) + t_2))) + (sqrt(y) - sqrt((1.0 + y))));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    if (y <= 4.5d-27) then
        tmp = ((t_2 - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + 2.0d0
    else if (y <= 3.3d+28) then
        tmp = t_1 - ((sqrt(x) + ((-1.0d0) / (sqrt(z) + t_2))) + (sqrt(y) - sqrt((1.0d0 + y))))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 4.5e-27) {
		tmp = ((t_2 - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + 2.0;
	} else if (y <= 3.3e+28) {
		tmp = t_1 - ((Math.sqrt(x) + (-1.0 / (Math.sqrt(z) + t_2))) + (Math.sqrt(y) - Math.sqrt((1.0 + y))));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 4.5e-27:
		tmp = ((t_2 - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + 2.0
	elif y <= 3.3e+28:
		tmp = t_1 - ((math.sqrt(x) + (-1.0 / (math.sqrt(z) + t_2))) + (math.sqrt(y) - math.sqrt((1.0 + y))))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 4.5e-27)
		tmp = Float64(Float64(Float64(t_2 - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + 2.0);
	elseif (y <= 3.3e+28)
		tmp = Float64(t_1 - Float64(Float64(sqrt(x) + Float64(-1.0 / Float64(sqrt(z) + t_2))) + Float64(sqrt(y) - sqrt(Float64(1.0 + y)))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 4.5e-27)
		tmp = ((t_2 - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	elseif (y <= 3.3e+28)
		tmp = t_1 - ((sqrt(x) + (-1.0 / (sqrt(z) + t_2))) + (sqrt(y) - sqrt((1.0 + y))));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.5e-27], N[(N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 3.3e+28], N[(t$95$1 - N[(N[(N[Sqrt[x], $MachinePrecision] + N[(-1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.5000000000000002e-27

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 59.8%

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

      3. +-commutative 59.8%

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

      4. sub-neg 59.8%

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

      5. sub-neg 59.8%

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

      6. +-commutative 59.8%

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

      7. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in x around 0 54.9%

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

   5. Taylor expanded in y around 0 54.9%

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

   if 4.5000000000000002e-27 < y < 3.3e28

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. +-commutative 88.9%

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

      3. associate-+r- 69.2%

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

      4. associate-+l- 49.8%

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

      5. +-commutative 49.8%

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

      6. associate--l+ 49.8%

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

      7. +-commutative 49.8%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in t around inf 35.5%

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

      1. +-commutative 35.5%

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

      2. +-commutative 35.5%

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

      3. associate--l+ 35.5%

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

   6. Simplified 35.5%

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

      1. flip-- 35.5%

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

      2. add-sqr-sqrt 22.7%

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

      3. add-sqr-sqrt 35.5%

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

      4. +-commutative 35.5%

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

      5. +-commutative 35.5%

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

   8. Applied egg-rr 35.5%

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

      1. associate--r+ 35.5%

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

      2. +-commutative 35.5%

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

   10. Simplified 35.5%

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

   11. Taylor expanded in z around 0 35.5%

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

   if 3.3e28 < y

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 87.8%

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

      4. associate-+l- 55.8%

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

      5. +-commutative 55.8%

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

      6. associate--l+ 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 34.0%

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

      1. +-commutative 34.0%

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

      2. +-commutative 34.0%

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

      3. associate--l+ 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in z around inf 18.1%

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

      1. +-commutative 18.1%

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

   9. Simplified 18.1%

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

   10. Taylor expanded in y around inf 18.1%

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

       1. flip-- 87.8%

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

       2. add-sqr-sqrt 74.6%

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

       3. +-commutative 74.6%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.4%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{1 + x} - \left(\left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 9: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\left(t_1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.5e-27)
     (+ (+ t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) 2.0)
     (if (<= y 3.3e+28)
       (+ t_2 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- t_1 (sqrt x))))
       (/ (+ 1.0 (- x x)) (+ (sqrt x) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.5e-27) {
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	} else if (y <= 3.3e+28) {
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) + (t_1 - sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_2);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.5d-27) then
        tmp = (t_1 + (sqrt((1.0d0 + t)) - sqrt(t))) + 2.0d0
    else if (y <= 3.3d+28) then
        tmp = t_2 + ((sqrt((1.0d0 + y)) - sqrt(y)) + (t_1 - sqrt(x)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.5e-27) {
		tmp = (t_1 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + 2.0;
	} else if (y <= 3.3e+28) {
		tmp = t_2 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (t_1 - Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.5e-27:
		tmp = (t_1 + (math.sqrt((1.0 + t)) - math.sqrt(t))) + 2.0
	elif y <= 3.3e+28:
		tmp = t_2 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (t_1 - math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + t_2)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.5e-27)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + 2.0);
	elseif (y <= 3.3e+28)
		tmp = Float64(t_2 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(t_1 - sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.5e-27)
		tmp = (t_1 + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	elseif (y <= 3.3e+28)
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) + (t_1 - sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.5e-27], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 3.3e+28], N[(t$95$2 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.5000000000000002e-27

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 59.8%

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

      3. +-commutative 59.8%

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

      4. sub-neg 59.8%

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

      5. sub-neg 59.8%

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

      6. +-commutative 59.8%

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

      7. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in x around 0 54.9%

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

   5. Taylor expanded in y around 0 54.9%

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

   if 4.5000000000000002e-27 < y < 3.3e28

   1. Initial program 88.9%

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

      1. associate-+l+ 88.9%

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

      2. +-commutative 88.9%

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

      3. associate-+r- 69.2%

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

      4. associate-+l- 49.8%

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

      5. +-commutative 49.8%

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

      6. associate--l+ 49.8%

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

      7. +-commutative 49.8%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in t around inf 35.5%

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

      1. +-commutative 35.5%

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

      2. +-commutative 35.5%

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

      3. associate--l+ 35.5%

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

   6. Simplified 35.5%

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

   if 3.3e28 < y

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 87.8%

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

      4. associate-+l- 55.8%

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

      5. +-commutative 55.8%

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

      6. associate--l+ 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 34.0%

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

      1. +-commutative 34.0%

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

      2. +-commutative 34.0%

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

      3. associate--l+ 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in z around inf 18.1%

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

      1. +-commutative 18.1%

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

   9. Simplified 18.1%

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

   10. Taylor expanded in y around inf 18.1%

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

       1. flip-- 87.8%

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

       2. add-sqr-sqrt 74.6%

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

       3. +-commutative 74.6%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.4%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 10: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+28)
   (+
    (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t)))
    (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))
   (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+28) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (sqrt((1.0 + y)) - sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.3d+28) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + (1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+28) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.3e+28:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+28)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.3e+28)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + (sqrt((1.0 + y)) - sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+28], N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e28

   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- 60.6%

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

      3. +-commutative 60.6%

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

      4. sub-neg 60.6%

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

      5. sub-neg 60.6%

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

      6. +-commutative 60.6%

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

      7. +-commutative 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. associate--l+ 55.0%

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

   6. Simplified 55.0%

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

   if 3.3e28 < y

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 87.8%

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

      4. associate-+l- 55.8%

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

      5. +-commutative 55.8%

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

      6. associate--l+ 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 34.0%

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

      1. +-commutative 34.0%

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

      2. +-commutative 34.0%

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

      3. associate--l+ 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in z around inf 18.1%

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

      1. +-commutative 18.1%

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

   9. Simplified 18.1%

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

   10. Taylor expanded in y around inf 18.1%

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

       1. flip-- 87.8%

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

       2. add-sqr-sqrt 74.6%

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

       3. +-commutative 74.6%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.4%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.9%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11: 90.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+28)
   (+ 1.0 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt z) (sqrt (+ 1.0 z)))))
   (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+28) {
		tmp = 1.0 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(z) - sqrt((1.0 + z))));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.3d+28) then
        tmp = 1.0d0 + ((sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(z) - sqrt((1.0d0 + z))))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+28) {
		tmp = 1.0 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(z) - Math.sqrt((1.0 + z))));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.3e+28:
		tmp = 1.0 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(z) - math.sqrt((1.0 + z))))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+28)
		tmp = Float64(1.0 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(z) - sqrt(Float64(1.0 + z)))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.3e+28)
		tmp = 1.0 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(z) - sqrt((1.0 + z))));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+28], N[(1.0 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e28

   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. +-commutative 97.5%

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

      3. associate-+r- 60.6%

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

      4. associate-+l- 53.8%

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

      5. +-commutative 53.8%

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

      6. associate--l+ 53.8%

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

      7. +-commutative 53.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 34.0%

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

      1. +-commutative 34.0%

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

      2. +-commutative 34.0%

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

      3. associate--l+ 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 34.0%

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

      1. associate--l+ 52.4%

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

      2. associate--l+ 52.4%

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

      3. +-commutative 52.4%

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

   9. Simplified 52.4%

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

       1. associate-+r- 52.4%

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

   11. Applied egg-rr 52.4%

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

       1. associate-+r- 52.4%

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

       2. +-commutative 52.4%

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

       3. associate--r+ 52.4%

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

       4. associate-+l- 52.1%

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

   13. Simplified 52.1%

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

   if 3.3e28 < y

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 87.8%

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

      4. associate-+l- 55.8%

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

      5. +-commutative 55.8%

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

      6. associate--l+ 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 34.0%

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

      1. +-commutative 34.0%

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

      2. +-commutative 34.0%

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

      3. associate--l+ 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in z around inf 18.1%

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

      1. +-commutative 18.1%

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

   9. Simplified 18.1%

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

   10. Taylor expanded in y around inf 18.1%

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

       1. flip-- 87.8%

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

       2. add-sqr-sqrt 74.6%

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

       3. +-commutative 74.6%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.4%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12: 95.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.5e-24)
   (+ (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (sqrt (+ 1.0 t)) (sqrt t))) 2.0)
   (if (<= y 2e+21)
     (- (+ 1.0 (* x 0.5)) (- (+ (sqrt y) (sqrt x)) (sqrt (+ 1.0 y))))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.5e-24) {
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	} else if (y <= 2e+21) {
		tmp = (1.0 + (x * 0.5)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0 + y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 7.5d-24) then
        tmp = ((sqrt((1.0d0 + z)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t))) + 2.0d0
    else if (y <= 2d+21) then
        tmp = (1.0d0 + (x * 0.5d0)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0d0 + y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.5e-24) {
		tmp = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + 2.0;
	} else if (y <= 2e+21) {
		tmp = (1.0 + (x * 0.5)) - ((Math.sqrt(y) + Math.sqrt(x)) - Math.sqrt((1.0 + y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 7.5e-24:
		tmp = ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + 2.0
	elif y <= 2e+21:
		tmp = (1.0 + (x * 0.5)) - ((math.sqrt(y) + math.sqrt(x)) - math.sqrt((1.0 + y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.5e-24)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + 2.0);
	elseif (y <= 2e+21)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - Float64(Float64(sqrt(y) + sqrt(x)) - sqrt(Float64(1.0 + y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7.5e-24)
		tmp = ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t))) + 2.0;
	elseif (y <= 2e+21)
		tmp = (1.0 + (x * 0.5)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0 + y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 7.5e-24], N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 2e+21], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.50000000000000007e-24

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l- 60.1%

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

      3. +-commutative 60.1%

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

      4. sub-neg 60.1%

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

      5. sub-neg 60.1%

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

      6. +-commutative 60.1%

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

      7. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 55.1%

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

   5. Taylor expanded in y around 0 55.1%

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

   if 7.50000000000000007e-24 < y < 2e21

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 64.5%

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

      4. associate-+l- 49.5%

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

      5. +-commutative 49.5%

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

      6. associate--l+ 49.5%

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

      7. +-commutative 49.5%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in t around inf 39.6%

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

      1. +-commutative 39.6%

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

      2. +-commutative 39.6%

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

      3. associate--l+ 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in z around inf 32.3%

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

      1. +-commutative 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in x around 0 32.9%

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

   if 2e21 < y

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-+r- 87.9%

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

      4. associate-+l- 55.4%

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

      5. +-commutative 55.4%

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

      6. associate--l+ 55.4%

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

      7. +-commutative 55.4%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around inf 33.7%

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

      1. +-commutative 33.7%

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

      2. +-commutative 33.7%

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

      3. associate--l+ 34.4%

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

   6. Simplified 34.4%

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

   7. Taylor expanded in z around inf 17.9%

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

      1. +-commutative 17.9%

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

   9. Simplified 17.9%

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

   10. Taylor expanded in y around inf 18.0%

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

       1. flip-- 87.9%

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

       2. add-sqr-sqrt 73.9%

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

       3. +-commutative 73.9%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.3%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.7%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13: 89.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 8.5e-24)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (if (<= y 2e+21)
     (- (+ 1.0 (* x 0.5)) (- (+ (sqrt y) (sqrt x)) (sqrt (+ 1.0 y))))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.5e-24) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 2e+21) {
		tmp = (1.0 + (x * 0.5)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0 + y)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 8.5d-24) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else if (y <= 2d+21) then
        tmp = (1.0d0 + (x * 0.5d0)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0d0 + y)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.5e-24) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 2e+21) {
		tmp = (1.0 + (x * 0.5)) - ((Math.sqrt(y) + Math.sqrt(x)) - Math.sqrt((1.0 + y)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 8.5e-24:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 2e+21:
		tmp = (1.0 + (x * 0.5)) - ((math.sqrt(y) + math.sqrt(x)) - math.sqrt((1.0 + y)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8.5e-24)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 2e+21)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - Float64(Float64(sqrt(y) + sqrt(x)) - sqrt(Float64(1.0 + y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8.5e-24)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	elseif (y <= 2e+21)
		tmp = (1.0 + (x * 0.5)) - ((sqrt(y) + sqrt(x)) - sqrt((1.0 + y)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 8.5e-24], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 2e+21], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5000000000000002e-24

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r- 60.0%

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

      4. associate-+l- 54.5%

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

      5. +-commutative 54.5%

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

      6. associate--l+ 54.5%

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

      7. +-commutative 54.5%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in t around inf 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. associate--l+ 35.6%

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

   6. Simplified 35.6%

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

   7. Taylor expanded in x around 0 34.0%

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

      1. associate--l+ 52.6%

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

      2. associate--l+ 52.6%

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

      3. +-commutative 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in y around 0 34.0%

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

       1. associate--l+ 51.3%

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

   12. Simplified 51.3%

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

   if 8.5000000000000002e-24 < y < 2e21

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 64.5%

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

      4. associate-+l- 49.5%

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

      5. +-commutative 49.5%

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

      6. associate--l+ 49.5%

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

      7. +-commutative 49.5%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in t around inf 39.6%

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

      1. +-commutative 39.6%

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

      2. +-commutative 39.6%

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

      3. associate--l+ 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in z around inf 32.3%

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

      1. +-commutative 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in x around 0 32.9%

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

   if 2e21 < y

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-+r- 87.9%

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

      4. associate-+l- 55.4%

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

      5. +-commutative 55.4%

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

      6. associate--l+ 55.4%

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

      7. +-commutative 55.4%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around inf 33.7%

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

      1. +-commutative 33.7%

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

      2. +-commutative 33.7%

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

      3. associate--l+ 34.4%

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

   6. Simplified 34.4%

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

   7. Taylor expanded in z around inf 17.9%

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

      1. +-commutative 17.9%

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

   9. Simplified 17.9%

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

   10. Taylor expanded in y around inf 18.0%

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

       1. flip-- 87.9%

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

       2. add-sqr-sqrt 73.9%

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

       3. +-commutative 73.9%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.3%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.7%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 14: 89.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 7e-24)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (if (<= y 2e+21)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (+ (sqrt y) (sqrt x))))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7e-24) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 2e+21) {
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 7d-24) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else if (y <= 2d+21) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7e-24) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 2e+21) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 7e-24:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 2e+21:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7e-24)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 2e+21)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7e-24)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	elseif (y <= 2e+21)
		tmp = 1.0 + (sqrt((1.0 + y)) - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 7e-24], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 2e+21], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.9999999999999993e-24

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+r- 60.0%

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

      4. associate-+l- 54.5%

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

      5. +-commutative 54.5%

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

      6. associate--l+ 54.5%

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

      7. +-commutative 54.5%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in t around inf 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. associate--l+ 35.6%

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

   6. Simplified 35.6%

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

   7. Taylor expanded in x around 0 34.0%

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

      1. associate--l+ 52.6%

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

      2. associate--l+ 52.6%

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

      3. +-commutative 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in y around 0 34.0%

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

       1. associate--l+ 51.3%

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

   12. Simplified 51.3%

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

   if 6.9999999999999993e-24 < y < 2e21

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 64.5%

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

      4. associate-+l- 49.5%

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

      5. +-commutative 49.5%

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

      6. associate--l+ 49.5%

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

      7. +-commutative 49.5%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in t around inf 39.6%

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

      1. +-commutative 39.6%

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

      2. +-commutative 39.6%

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

      3. associate--l+ 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in z around inf 32.3%

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

      1. +-commutative 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in x around 0 31.4%

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

   if 2e21 < y

   1. Initial program 87.9%

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

      1. associate-+l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-+r- 87.9%

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

      4. associate-+l- 55.4%

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

      5. +-commutative 55.4%

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

      6. associate--l+ 55.4%

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

      7. +-commutative 55.4%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around inf 33.7%

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

      1. +-commutative 33.7%

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

      2. +-commutative 33.7%

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

      3. associate--l+ 34.4%

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

   6. Simplified 34.4%

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

   7. Taylor expanded in z around inf 17.9%

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

      1. +-commutative 17.9%

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

   9. Simplified 17.9%

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

   10. Taylor expanded in y around inf 18.0%

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

       1. flip-- 87.9%

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

       2. add-sqr-sqrt 73.9%

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

       3. +-commutative 73.9%

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

       4. add-sqr-sqrt 88.4%

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

       5. +-commutative 88.4%

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

   12. Applied egg-rr 18.3%

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

       1. +-commutative 88.4%

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

       2. associate--l+ 90.8%

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

       3. +-commutative 90.8%

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

       4. +-commutative 90.8%

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

   14. Simplified 21.7%

       \[\leadsto \color{blue}{\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 15: 89.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+22}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.6e-24)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (if (<= y 1e+22)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt (+ 1.0 x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-24) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 1e+22) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.6d-24) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else if (y <= 1d+22) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-24) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 1e+22) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 4.6e-24:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 1e+22:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.6e-24)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 1e+22)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.6e-24)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	elseif (y <= 1e+22)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 4.6e-24], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1e+22], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.6000000000000002e-24

   1. Initial program 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 60.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 54.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 54.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 40.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.8%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 34.0%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 52.6%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

      2. associate--l+ 52.6%

         \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

      3. +-commutative 52.6%

         \[\leadsto 1 + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   9. Simplified 52.6%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. Taylor expanded in y around 0 34.0%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   11. Step-by-step derivation

       1. associate--l+ 51.3%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   12. Simplified 51.3%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 4.6000000000000002e-24 < y < 1e22

   1. Initial program 87.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 87.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 87.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 64.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 49.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 49.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 49.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 49.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 31.4%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 39.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 39.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 39.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 39.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 39.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 32.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 32.3%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 32.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 53.2%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 53.2%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 53.2%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 1e22 < y

   1. Initial program 87.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 87.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 87.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 87.9%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 55.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 55.4%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 55.4%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 55.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 33.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 33.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 34.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 34.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 17.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 17.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 17.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 18.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 87.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       2. add-sqr-sqrt 73.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. +-commutative 73.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       4. add-sqr-sqrt 88.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       5. +-commutative 88.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   12. Applied egg-rr 18.3%

       \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} \]
   13. Step-by-step derivation

       1. +-commutative 88.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       2. associate--l+ 90.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       3. +-commutative 90.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

       4. +-commutative 90.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   14. Simplified 21.7%

       \[\leadsto \color{blue}{\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+22}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 16: 61.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.15:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 x)) (sqrt x)))) (if (<= y 1.15) (+ 1.0 t_1) t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (y <= 1.15) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x)) - sqrt(x)
    if (y <= 1.15d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (y <= 1.15) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if y <= 1.15:
		tmp = 1.0 + t_1
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (y <= 1.15)
		tmp = Float64(1.0 + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (y <= 1.15)
		tmp = 1.0 + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.15], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 60.3%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 54.7%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 54.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 40.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 34.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 34.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 34.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 36.0%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 36.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 20.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 20.3%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 20.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around 0 20.2%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate--l+ 37.7%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   12. Simplified 37.7%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   if 1.1499999999999999 < y

   1. Initial program 87.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 87.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 87.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 86.5%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 54.6%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 54.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.8%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 34.4%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 34.4%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 18.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 18.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in y around inf 18.5%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 17: 84.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.6e-24)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-24) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.6d-24) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e-24) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 4.6e-24:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.6e-24)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.6e-24)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 4.6e-24], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.6000000000000002e-24

   1. Initial program 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 98.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 98.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 60.0%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 54.5%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 54.5%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 40.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 33.8%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 33.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 35.6%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 35.6%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in x around 0 34.0%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 52.6%

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

      2. associate--l+ 52.6%

         \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)} \]

      3. +-commutative 52.6%

         \[\leadsto 1 + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   9. Simplified 52.6%

      \[\leadsto \color{blue}{1 + \left(\sqrt{1 + z} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. Taylor expanded in y around 0 34.0%

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \sqrt{z}} \]
   11. Step-by-step derivation

       1. associate--l+ 51.3%

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   12. Simplified 51.3%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 4.6000000000000002e-24 < y

   1. Initial program 87.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 87.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 87.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 85.7%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 54.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 54.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. associate--l+ 54.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

      7. +-commutative 54.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 34.2%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
   5. Step-by-step derivation

      1. +-commutative 34.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

      2. +-commutative 34.2%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

      3. associate--l+ 34.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   6. Simplified 34.8%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   7. Taylor expanded in z around inf 19.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 19.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   9. Simplified 19.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   10. Taylor expanded in x around 0 9.0%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   11. Step-by-step derivation

       1. associate--l+ 43.6%

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   12. Simplified 43.6%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18: 64.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt((1.0 + y)) - sqrt(y));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 93.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 72.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 54.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 54.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. associate--l+ 54.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

   7. +-commutative 54.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

3. Simplified 39.9%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

4. Taylor expanded in t around inf 34.0%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 34.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

   2. +-commutative 34.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

   3. associate--l+ 35.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

6. Simplified 35.3%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 19.6%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 19.6%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 19.6%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in x around 0 26.6%

    \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
11. Step-by-step derivation

    1. associate--l+ 43.0%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

12. Simplified 43.0%

    \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

13. Final simplification 43.0%

    \[\leadsto 1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \]

Alternative 19: 36.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{1 + x} - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((1.0 + x)) - sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((1.0d0 + x)) - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((1.0 + x)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((1.0 + x)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((1.0 + x)) - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 93.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 93.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 72.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 54.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 54.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. associate--l+ 54.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]

   7. +-commutative 54.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

3. Simplified 39.9%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) + \left(\sqrt{t} + \left(\sqrt{z} - \sqrt{1 + t}\right)\right)\right)\right)} \]

4. Taylor expanded in t around inf 34.0%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 34.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]

   2. +-commutative 34.0%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]

   3. associate--l+ 35.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

6. Simplified 35.3%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

7. Taylor expanded in z around inf 19.6%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 19.6%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

9. Simplified 19.6%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

10. Taylor expanded in y around inf 14.2%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Final simplification 14.2%

    \[\leadsto \sqrt{1 + x} - \sqrt{x} \]

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 2023199 
(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))))