Main:z from

Percentage Accurate: 91.8% → 98.9%

Time: 26.3s

Precision: binary64

Cost: 125192

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := 1 + \left(t + t\right)\\ t_4 := \sqrt{1 + y}\\ t_5 := t_2 + \left(\left(t_4 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t_5 \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{elif}\;t_5 \leq 2.2:\\ \;\;\;\;t_1 + \left(\left(\frac{1}{\sqrt{z} + \mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{x}\right) - \frac{-1}{t_4 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t_4\right) - \sqrt{y}\right) + \left(t_2 + \frac{\frac{t_3}{t_3}}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \end{array} \]

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))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (+ 1.0 (+ t t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (+ t_2 (+ (- t_4 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_5 0.1)
     (/ 1.0 (+ (sqrt x) t_1))
     (if (<= t_5 2.2)
       (+
        t_1
        (-
         (- (/ 1.0 (+ (sqrt z) (hypot 1.0 (sqrt z)))) (sqrt x))
         (/ -1.0 (+ t_4 (sqrt y)))))
       (+
        (- (+ 1.0 t_4) (sqrt y))
        (+ t_2 (/ (/ t_3 t_3) (+ (sqrt t) (sqrt (+ 1.0 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));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = 1.0 + (t + t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = t_2 + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_5 <= 0.1) {
		tmp = 1.0 / (sqrt(x) + t_1);
	} else if (t_5 <= 2.2) {
		tmp = t_1 + (((1.0 / (sqrt(z) + hypot(1.0, sqrt(z)))) - sqrt(x)) - (-1.0 / (t_4 + sqrt(y))));
	} else {
		tmp = ((1.0 + t_4) - sqrt(y)) + (t_2 + ((t_3 / t_3) / (sqrt(t) + sqrt((1.0 + t)))));
	}
	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((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_3 = 1.0 + (t + t);
	double t_4 = Math.sqrt((1.0 + y));
	double t_5 = t_2 + ((t_4 - Math.sqrt(y)) + (t_1 - Math.sqrt(x)));
	double tmp;
	if (t_5 <= 0.1) {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	} else if (t_5 <= 2.2) {
		tmp = t_1 + (((1.0 / (Math.sqrt(z) + Math.hypot(1.0, Math.sqrt(z)))) - Math.sqrt(x)) - (-1.0 / (t_4 + Math.sqrt(y))));
	} else {
		tmp = ((1.0 + t_4) - Math.sqrt(y)) + (t_2 + ((t_3 / t_3) / (Math.sqrt(t) + Math.sqrt((1.0 + t)))));
	}
	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((1.0 + x))
	t_2 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_3 = 1.0 + (t + t)
	t_4 = math.sqrt((1.0 + y))
	t_5 = t_2 + ((t_4 - math.sqrt(y)) + (t_1 - math.sqrt(x)))
	tmp = 0
	if t_5 <= 0.1:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	elif t_5 <= 2.2:
		tmp = t_1 + (((1.0 / (math.sqrt(z) + math.hypot(1.0, math.sqrt(z)))) - math.sqrt(x)) - (-1.0 / (t_4 + math.sqrt(y))))
	else:
		tmp = ((1.0 + t_4) - math.sqrt(y)) + (t_2 + ((t_3 / t_3) / (math.sqrt(t) + math.sqrt((1.0 + t)))))
	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(1.0 + x))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = Float64(1.0 + Float64(t + t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = Float64(t_2 + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_5 <= 0.1)
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	elseif (t_5 <= 2.2)
		tmp = Float64(t_1 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + hypot(1.0, sqrt(z)))) - sqrt(x)) - Float64(-1.0 / Float64(t_4 + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(1.0 + t_4) - sqrt(y)) + Float64(t_2 + Float64(Float64(t_3 / t_3) / Float64(sqrt(t) + sqrt(Float64(1.0 + t))))));
	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((1.0 + x));
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	t_3 = 1.0 + (t + t);
	t_4 = sqrt((1.0 + y));
	t_5 = t_2 + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	tmp = 0.0;
	if (t_5 <= 0.1)
		tmp = 1.0 / (sqrt(x) + t_1);
	elseif (t_5 <= 2.2)
		tmp = t_1 + (((1.0 / (sqrt(z) + hypot(1.0, sqrt(z)))) - sqrt(x)) - (-1.0 / (t_4 + sqrt(y))));
	else
		tmp = ((1.0 + t_4) - sqrt(y)) + (t_2 + ((t_3 / t_3) / (sqrt(t) + sqrt((1.0 + t)))));
	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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.1], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.2], N[(t$95$1 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$4), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(t$95$3 / t$95$3), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	91.8%
Target	99.3%
Herbie	98.9%

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

Derivation

1. Split input into 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))) < 0.10000000000000001

   1. Initial program 50.0%

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

   2. Simplified 5.1%

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

      [Start] 50.0%	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 50.0%	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 50.0%	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 48.0%	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 4.4%	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 4.4%	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 4.4%	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 4.4%	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 4.7%

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

   4. Simplified 4.4%

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

      [Start] 4.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 4.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 4.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]
      associate--l+ [=>] 4.4%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   5. Taylor expanded in z around inf 5.0%

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

   6. Simplified 5.0%

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

      [Start] 5.0%	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
      +-commutative [=>] 5.0%	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Taylor expanded in y around inf 4.4%

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

   8. Applied egg-rr 6.5%

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

      [Start] 4.4%	\[ \sqrt{1 + x} - \sqrt{x} \]
      flip-- [=>] 4.4%	\[ \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]
      add-sqr-sqrt [<=] 4.3%	\[ \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [<=] 4.3%	\[ \frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \]
      add-sqr-sqrt [<=] 6.5%	\[ \frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [<=] 6.5%	\[ \frac{\left(x + 1\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} \]

   9. Simplified 21.8%

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

      [Start] 6.5%	\[ \frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} \]
      +-commutative [=>] 6.5%	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt{x + 1} + \sqrt{x}} \]
      associate--l+ [=>] 21.8%	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
      +-inverses [=>] 21.8%	\[ \frac{1 + \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]
      metadata-eval [=>] 21.8%	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
      +-commutative [=>] 21.8%	\[ \frac{1}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} \]
      +-commutative [=>] 21.8%	\[ \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} \]

   if 0.10000000000000001 < (+.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))) < 2.2000000000000002

   1. Initial program 96.6%

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

   2. Simplified 35.1%

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

      [Start] 96.6%	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 96.6%	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 96.6%	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 71.8%	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 52.6%	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 52.6%	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 52.6%	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 52.6%	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 26.7%

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

   4. Simplified 27.0%

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

      [Start] 26.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 26.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 26.7%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{z}\right) - \sqrt{\color{blue}{z + 1}}\right)\right) \]
      associate--l+ [=>] 27.0%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)}\right) \]

   5. Applied egg-rr 27.1%

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

      [Start] 27.0%	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      flip-- [=>] 27.1%	\[ \sqrt{x + 1} + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      add-sqr-sqrt [<=] 21.9%	\[ \sqrt{x + 1} + \left(\frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      add-sqr-sqrt [<=] 27.1%	\[ \sqrt{x + 1} + \left(\frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   6. Simplified 27.6%

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

      [Start] 27.1%	\[ \sqrt{x + 1} + \left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      associate--l+ [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      +-inverses [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      metadata-eval [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{\color{blue}{1}}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]

   7. Applied egg-rr 27.6%

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

      [Start] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\right) \]
      flip-- [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{z} \cdot \sqrt{z} - \sqrt{z + 1} \cdot \sqrt{z + 1}}{\sqrt{z} + \sqrt{z + 1}}}\right)\right) \]
      add-sqr-sqrt [<=] 21.1%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{\color{blue}{z} - \sqrt{z + 1} \cdot \sqrt{z + 1}}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      add-sqr-sqrt [<=] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{z - \color{blue}{\left(z + 1\right)}}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      +-commutative [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{z - \color{blue}{\left(1 + z\right)}}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      +-commutative [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{z - \left(1 + z\right)}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right)\right) \]

   8. Simplified 27.8%

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

      [Start] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{z - \left(1 + z\right)}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      +-commutative [=>] 27.6%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{z - \color{blue}{\left(z + 1\right)}}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      associate--r+ [=>] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{\color{blue}{\left(z - z\right) - 1}}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      +-inverses [=>] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{\color{blue}{0} - 1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      metadata-eval [=>] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{\color{blue}{-1}}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      +-commutative [=>] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{-1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) \]
      rem-square-sqrt [<=] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{-1}{\sqrt{1 + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} + \sqrt{z}}\right)\right) \]
      hypot-1-def [=>] 27.8%	\[ \sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{x} + \frac{-1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{z}\right)} + \sqrt{z}}\right)\right) \]

   if 2.2000000000000002 < (+.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 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0%	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 100.0%	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 100.0%	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      add-sqr-sqrt [<=] 75.9%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 75.9%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\left(t + 1\right) - t}{\sqrt{\color{blue}{t + 1}} + \sqrt{t}}\right) \]

   4. Applied egg-rr 75.0%

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

      [Start] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      flip-- [=>] 75.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\color{blue}{\frac{\left(t + 1\right) \cdot \left(t + 1\right) - t \cdot t}{\left(t + 1\right) + t}}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-commutative [=>] 75.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\color{blue}{\left(1 + t\right)} \cdot \left(t + 1\right) - t \cdot t}{\left(t + 1\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-commutative [=>] 75.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(1 + t\right) \cdot \color{blue}{\left(1 + t\right)} - t \cdot t}{\left(t + 1\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-commutative [=>] 75.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(1 + t\right) \cdot \left(1 + t\right) - t \cdot t}{\color{blue}{\left(1 + t\right)} + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]

   5. Simplified 100.0%

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

      [Start] 75.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(1 + t\right) \cdot \left(1 + t\right) - t \cdot t}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      difference-of-squares [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\color{blue}{\left(\left(1 + t\right) + t\right) \cdot \left(\left(1 + t\right) - t\right)}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-commutative [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \left(\color{blue}{\left(t + 1\right)} - t\right)}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      associate-+r- [<=] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \color{blue}{\left(t + \left(1 - t\right)\right)}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      associate-+r- [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \color{blue}{\left(\left(t + 1\right) - t\right)}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-commutative [<=] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \left(\color{blue}{\left(1 + t\right)} - t\right)}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      associate--l+ [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      +-inverses [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \left(1 + \color{blue}{0}\right)}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      metadata-eval [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\left(\left(1 + t\right) + t\right) \cdot \color{blue}{1}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      *-rgt-identity [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\color{blue}{\left(1 + t\right) + t}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      associate-+l+ [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{\color{blue}{1 + \left(t + t\right)}}{\left(1 + t\right) + t}}{\sqrt{t + 1} + \sqrt{t}}\right) \]
      associate-+l+ [=>] 100.0%	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{1 + \left(t + t\right)}{\color{blue}{1 + \left(t + t\right)}}}{\sqrt{t + 1} + \sqrt{t}}\right) \]

   6. Taylor expanded in x around 0 97.1%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.2:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\frac{1}{\sqrt{z} + \mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{x}\right) - \frac{-1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\frac{1 + \left(t + t\right)}{1 + \left(t + t\right)}}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	125192

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := 1 + \left(t + t\right)\\ t_4 := \sqrt{1 + y}\\ t_5 := t_2 + \left(\left(t_4 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t_5 \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{elif}\;t_5 \leq 2.2:\\ \;\;\;\;t_1 + \left(\left(\frac{1}{\sqrt{z} + \mathsf{hypot}\left(1, \sqrt{z}\right)} - \sqrt{x}\right) - \frac{-1}{t_4 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t_4\right) - \sqrt{y}\right) + \left(t_2 + \frac{\frac{t_3}{t_3}}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	79556

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;\left(t_1 - \sqrt{y}\right) + \left(t_2 - \sqrt{x}\right) \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 + \left(\frac{1}{t_1 + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	65728

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

Alternative 4
Accuracy	95.8%
Cost	65600

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

Alternative 5
Accuracy	91.9%
Cost	52804

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	95.9%
Cost	40132

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 7
Accuracy	97.0%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-51}:\\ \;\;\;\;\left(t_2 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 8
Accuracy	95.7%
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 9
Accuracy	90.7%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left(t_1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{y}} + t_2\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 10
Accuracy	91.3%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11
Accuracy	89.5%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;1 + \left(t_1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(t_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12
Accuracy	89.4%
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13
Accuracy	61.5%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.9:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Accuracy	81.8%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.9:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 15
Accuracy	86.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.65:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 16
Accuracy	35.2%
Cost	13120

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

Alternative 17
Accuracy	34.7%
Cost	6848

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

Alternative 18
Accuracy	34.1%
Cost	6592

\[1 - \sqrt{x} \]

Reproduce

herbie shell --seed 2023165 
(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))))