Main:z from

Average Accuracy: 92.4% → 97.0%

Time: 33.3s

Precision: binary64

Cost: 59204

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;y \leq 1.88 \cdot 10^{+59}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{t_1 + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - {\left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)}^{2}\right)}{\sqrt{x} + \left(\mathsf{hypot}\left(1, \sqrt{x}\right) + \left(\sqrt{y} - t_1\right)\right)}\\ \end{array} \]

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))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<= y 1.88e+59)
     (+
      (+ (sqrt (+ x 1.0)) (- (/ (+ 1.0 (- y y)) (+ t_1 (sqrt y))) (sqrt x)))
      (+
       (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))
       (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))))
     (/
      (+ 1.0 (- x (pow (- (- t_1 (sqrt y)) (sqrt x)) 2.0)))
      (+ (sqrt x) (+ (hypot 1.0 (sqrt x)) (- (sqrt y) t_1)))))))

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));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (y <= 1.88e+59) {
		tmp = (sqrt((x + 1.0)) + (((1.0 + (y - y)) / (t_1 + sqrt(y))) - sqrt(x))) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
	} else {
		tmp = (1.0 + (x - pow(((t_1 - sqrt(y)) - sqrt(x)), 2.0))) / (sqrt(x) + (hypot(1.0, sqrt(x)) + (sqrt(y) - t_1)));
	}
	return tmp;
}

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));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double tmp;
	if (y <= 1.88e+59) {
		tmp = (Math.sqrt((x + 1.0)) + (((1.0 + (y - y)) / (t_1 + Math.sqrt(y))) - Math.sqrt(x))) + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))));
	} else {
		tmp = (1.0 + (x - Math.pow(((t_1 - Math.sqrt(y)) - Math.sqrt(x)), 2.0))) / (Math.sqrt(x) + (Math.hypot(1.0, Math.sqrt(x)) + (Math.sqrt(y) - t_1)));
	}
	return tmp;
}

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))

↓

def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	tmp = 0
	if y <= 1.88e+59:
		tmp = (math.sqrt((x + 1.0)) + (((1.0 + (y - y)) / (t_1 + math.sqrt(y))) - math.sqrt(x))) + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))))
	else:
		tmp = (1.0 + (x - math.pow(((t_1 - math.sqrt(y)) - math.sqrt(x)), 2.0))) / (math.sqrt(x) + (math.hypot(1.0, math.sqrt(x)) + (math.sqrt(y) - t_1)))
	return tmp

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

↓

function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.88e+59)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(Float64(Float64(1.0 + Float64(y - y)) / Float64(t_1 + sqrt(y))) - sqrt(x))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - (Float64(Float64(t_1 - sqrt(y)) - sqrt(x)) ^ 2.0))) / Float64(sqrt(x) + Float64(hypot(1.0, sqrt(x)) + Float64(sqrt(y) - t_1))));
	end
	return tmp
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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	tmp = 0.0;
	if (y <= 1.88e+59)
		tmp = (sqrt((x + 1.0)) + (((1.0 + (y - y)) / (t_1 + sqrt(y))) - sqrt(x))) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
	else
		tmp = (1.0 + (x - (((t_1 - sqrt(y)) - sqrt(x)) ^ 2.0))) / (sqrt(x) + (hypot(1.0, sqrt(x)) + (sqrt(y) - t_1)));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.88e+59], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - N[Power[N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	92.4%
Target	99.5%
Herbie	97.0%

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

Derivation

1. Split input into 2 regimes

2. if y < 1.87999999999999989e59

   1. Initial program 95.9%

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

   2. Simplified 95.9%

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

      [Start] 95.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 95.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 95.9	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 95.4	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 95.4	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 95.9	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 96.6%

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

      [Start] 95.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 96.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 96.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 92.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 92.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(y + 1\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(y + \left(1 - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   4. Simplified 97.3%

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

      [Start] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*r/ [=>] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-rgt-identity [=>] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [<=] 96.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 97.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 97.5%

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

      [Start] 97.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      div-inv [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      add-sqr-sqrt [<=] 54.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 97.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   6. Simplified 97.8%

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

      [Start] 97.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-commutative [=>] 97.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\left(1 + t\right) - t\right)}\right) \]
      associate--l+ [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}\right) \]
      distribute-rgt-in [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)}\right) \]
      +-inverses [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{0} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [<=] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{\left(z - z\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      distribute-rgt-out [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(1 + \left(z - z\right)\right)}\right) \]
      +-commutative [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(\left(z - z\right) + 1\right)}\right) \]
      +-inverses [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\color{blue}{0} + 1\right)\right) \]
      metadata-eval [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   7. Applied egg-rr 99.3%

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

      [Start] 97.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      flip-- [=>] 98.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      div-inv [=>] 98.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 72.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 98.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      associate--l+ [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   8. Simplified 99.3%

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

      [Start] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   if 1.87999999999999989e59 < y

   1. Initial program 76.6%

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

   2. Simplified 5.7%

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

      [Start] 76.6	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 76.6	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 76.6	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 76.6	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 76.6	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 76.6	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 76.6	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 76.6	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 72.4%

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

   4. Simplified 76.6%

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

      [Start] 72.4	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 72.4	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      associate--l+ [=>] 76.6	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   5. Taylor expanded in z around inf 72.3%

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

   6. Simplified 72.3%

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

      [Start] 72.3	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
      +-commutative [=>] 72.3	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{\frac{1 + \left(x - {\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}^{2}\right)}{\sqrt{x} + \left(\mathsf{hypot}\left(1, \sqrt{x}\right) - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)}} \]
      Proof

      [Start] 72.3	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]
      flip-+ [=>] 72.3	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \cdot \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)}} \]
      add-sqr-sqrt [<=] 72.3	\[ \frac{\color{blue}{\left(x + 1\right)} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \cdot \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
      +-commutative [=>] 72.3	\[ \frac{\color{blue}{\left(1 + x\right)} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \cdot \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
      associate--l+ [=>] 72.3	\[ \frac{\color{blue}{1 + \left(x - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \cdot \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)}}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
      pow2 [=>] 72.3	\[ \frac{1 + \left(x - \color{blue}{{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)}^{2}}\right)}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
      associate--r+ [=>] 74.7	\[ \frac{1 + \left(x - {\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}}^{2}\right)}{\sqrt{x + 1} - \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
      associate--r+ [=>] 86.9	\[ \frac{1 + \left(x - {\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}^{2}\right)}{\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.88 \cdot 10^{+59}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{\sqrt{y + 1} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - {\left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \sqrt{x}\right)}^{2}\right)}{\sqrt{x} + \left(\mathsf{hypot}\left(1, \sqrt{x}\right) + \left(\sqrt{y} - \sqrt{y + 1}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.3%
Cost	53312

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

Alternative 2
Accuracy	94.0%
Cost	53184

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

Alternative 3
Accuracy	94.1%
Cost	52992

\[\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{x + 1} - \left(\sqrt{x} + \left(\sqrt{y} - {\left(y + 1\right)}^{0.5}\right)\right)\right) \]

Alternative 4
Accuracy	94.9%
Cost	40132

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1} + \sqrt{y}\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{+63}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{t_1 + \sqrt{z}}\right) + \left(1 + \frac{1}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{t_2} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	93.9%
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;z \leq 1.2 \cdot 10^{+27}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_1\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{t_2 + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	93.7%
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{y + 1} + \sqrt{y}\\ \mathbf{if}\;z \leq 8 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_1\right) + \left(1 + \frac{1}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{t_2} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	94.7%
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{t_1 + \sqrt{z}}\right) + \left(\left(1 + t_2\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y + 1} + \sqrt{y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	93.8%
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t_1 - \sqrt{z}\right)\right) + \left(1 + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(t_2 + \left(\frac{1}{t_1 + \sqrt{z}} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	93.4%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 560000000000:\\ \;\;\;\;\left(\sqrt{1 + t} + \left(\left(t_1 + 2\right) - \sqrt{z}\right)\right) - \sqrt{t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\frac{1}{t_1 + \sqrt{z}} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	93.2%
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;2 + \left(\frac{1}{t_1 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	93.2%
Cost	26820

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 530000000000:\\ \;\;\;\;\left(\sqrt{1 + t} + \left(\left(t_1 + 2\right) - \sqrt{z}\right)\right) - \sqrt{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \left(1 + \frac{1}{t_1 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 12
Accuracy	91.8%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 8 \cdot 10^{-31}:\\ \;\;\;\;2 + \left(\frac{1}{t_1 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	91.0%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 8000000000000:\\ \;\;\;\;2 + \left(t_1 + \left(\left(\sqrt{1 + t} - \sqrt{z}\right) - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	91.0%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 530000000000:\\ \;\;\;\;\left(\sqrt{1 + t} + \left(\left(t_1 + 2\right) - \sqrt{z}\right)\right) - \sqrt{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	85.3%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	85.7%
Cost	26432

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

Alternative 17
Accuracy	85.3%
Cost	20164

\[\begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(1 + x \cdot 0.5\right)\\ \end{array} \]

Alternative 18
Accuracy	85.2%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 19
Accuracy	86.0%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{y + 1} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 20
Accuracy	82.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 21
Accuracy	35.7%
Cost	13120

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

Alternative 22
Accuracy	34.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(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))))