Main:z from

Average Accuracy: 92.0% → 98.7%

Time: 32.8s

Precision: binary64

Cost: 40012

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;\left(t_2 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + 2\\ \mathbf{elif}\;y \leq 28000:\\ \;\;\;\;t_3 + \left(\left(t_1 - \sqrt{y}\right) + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+32}:\\ \;\;\;\;t_3 + \left(1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_1} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3 + \sqrt{x}}\\ \end{array} \]

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 y)))
        (t_2 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
        (t_3 (sqrt (+ x 1.0))))
   (if (<= y 1.85e-31)
     (+ (+ t_2 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))) 2.0)
     (if (<= y 28000.0)
       (+ t_3 (+ (- t_1 (sqrt y)) (- t_2 (sqrt x))))
       (if (<= y 1e+32)
         (+
          t_3
          (+
           1.0
           (- (/ (+ 1.0 (- y y)) (+ (sqrt y) t_1)) (exp (log1p (sqrt x))))))
         (/ 1.0 (+ t_3 (sqrt x))))))))

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 + y));
	double t_2 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if (y <= 1.85e-31) {
		tmp = (t_2 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))) + 2.0;
	} else if (y <= 28000.0) {
		tmp = t_3 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)));
	} else if (y <= 1e+32) {
		tmp = t_3 + (1.0 + (((1.0 + (y - y)) / (sqrt(y) + t_1)) - exp(log1p(sqrt(x)))));
	} else {
		tmp = 1.0 / (t_3 + sqrt(x));
	}
	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 + y));
	double t_2 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double t_3 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 1.85e-31) {
		tmp = (t_2 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))) + 2.0;
	} else if (y <= 28000.0) {
		tmp = t_3 + ((t_1 - Math.sqrt(y)) + (t_2 - Math.sqrt(x)));
	} else if (y <= 1e+32) {
		tmp = t_3 + (1.0 + (((1.0 + (y - y)) / (Math.sqrt(y) + t_1)) - Math.exp(Math.log1p(Math.sqrt(x)))));
	} else {
		tmp = 1.0 / (t_3 + Math.sqrt(x));
	}
	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 + y))
	t_2 = 1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))
	t_3 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 1.85e-31:
		tmp = (t_2 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))) + 2.0
	elif y <= 28000.0:
		tmp = t_3 + ((t_1 - math.sqrt(y)) + (t_2 - math.sqrt(x)))
	elif y <= 1e+32:
		tmp = t_3 + (1.0 + (((1.0 + (y - y)) / (math.sqrt(y) + t_1)) - math.exp(math.log1p(math.sqrt(x)))))
	else:
		tmp = 1.0 / (t_3 + math.sqrt(x))
	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 + y))
	t_2 = Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.85e-31)
		tmp = Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))) + 2.0);
	elseif (y <= 28000.0)
		tmp = Float64(t_3 + Float64(Float64(t_1 - sqrt(y)) + Float64(t_2 - sqrt(x))));
	elseif (y <= 1e+32)
		tmp = Float64(t_3 + Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(y) + t_1)) - exp(log1p(sqrt(x))))));
	else
		tmp = Float64(1.0 / Float64(t_3 + sqrt(x)));
	end
	return 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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.85e-31], N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 28000.0], N[(t$95$3 + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+32], N[(t$95$3 + N[(1.0 + N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Exp[N[Log[1 + N[Sqrt[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	92.0%
Target	99.4%
Herbie	98.7%

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

Derivation

1. Split input into 4 regimes

2. if y < 1.8499999999999999e-31

   1. Initial program 97.7%

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

   2. Simplified 97.7%

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

      [Start] 97.7	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 97.7	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 97.7	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 97.7	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 97.7	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 97.7	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Taylor expanded in x around 0 97.7%

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

   4. Simplified 97.7%

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

      [Start] 97.7	\[ \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 97.7	\[ \color{blue}{\left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 98.0%

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

      [Start] 97.7	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 97.8	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      div-inv [=>] 97.8	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      add-sqr-sqrt [<=] 57.2	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 98.0	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   6. Simplified 98.3%

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

      [Start] 98.0	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-commutative [=>] 98.0	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\left(1 + t\right) - t\right)}\right) \]
      associate--l+ [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}\right) \]
      distribute-rgt-in [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)}\right) \]
      +-inverses [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{0} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [<=] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{\left(z - z\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      distribute-rgt-out [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(1 + \left(z - z\right)\right)}\right) \]
      +-commutative [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(\left(z - z\right) + 1\right)}\right) \]
      +-inverses [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\color{blue}{0} + 1\right)\right) \]
      metadata-eval [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   7. Applied egg-rr 99.9%

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

      [Start] 98.3	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      flip-- [=>] 98.6	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      div-inv [=>] 98.6	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 78.2	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 99.2	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      associate--l+ [=>] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   8. Simplified 99.9%

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

      [Start] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 99.9	\[ \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   9. Taylor expanded in y around 0 99.9%

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

   if 1.8499999999999999e-31 < y < 28000

   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 12.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + z}\right) - \left(\sqrt{1 + t} - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\right)} \]
      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	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 95.9	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 95.9	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 95.9	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 95.9	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 95.9	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 95.9	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 93.7%

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

   4. Simplified 95.4%

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

      [Start] 93.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 93.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      associate--l+ [=>] 95.4	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   5. Applied egg-rr 98.1%

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

      [Start] 95.4	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right) \]
      flip-- [=>] 95.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{1 + z}}{\sqrt{z} + \sqrt{1 + z}}}\right)\right) \]
      div-inv [=>] 95.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{1 + z}\right) \cdot \frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right)\right) \]
      add-sqr-sqrt [<=] 60.0	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\color{blue}{z} - \sqrt{1 + z} \cdot \sqrt{1 + z}\right) \cdot \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      add-sqr-sqrt [<=] 96.6	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(z - \color{blue}{\left(1 + z\right)}\right) \cdot \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      +-commutative [=>] 96.6	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(z - \color{blue}{\left(z + 1\right)}\right) \cdot \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      associate--r+ [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \color{blue}{\left(\left(z - z\right) - 1\right)} \cdot \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) \]
      +-commutative [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\left(z - z\right) - 1\right) \cdot \frac{1}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}}\right)\right) \]

   6. Simplified 98.1%

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

      [Start] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\left(z - z\right) - 1\right) \cdot \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      associate-*r/ [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \color{blue}{\frac{\left(\left(z - z\right) - 1\right) \cdot 1}{\sqrt{z} + \sqrt{z + 1}}}\right)\right) \]
      sub-neg [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{\color{blue}{\left(\left(z - z\right) + \left(-1\right)\right)} \cdot 1}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      +-inverses [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{\left(\color{blue}{0} + \left(-1\right)\right) \cdot 1}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      metadata-eval [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{\left(0 + \color{blue}{-1}\right) \cdot 1}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      metadata-eval [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{\color{blue}{-1} \cdot 1}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      metadata-eval [=>] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{\color{blue}{-1}}{\sqrt{z} + \sqrt{z + 1}}\right)\right) \]
      +-commutative [<=] 98.1	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \frac{-1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right)\right) \]

   if 28000 < y < 1.00000000000000005e32

   1. Initial program 78.7%

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

   2. Simplified 7.6%

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

      [Start] 78.7	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 78.7	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 78.7	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 78.7	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 78.7	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 78.7	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 78.7	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 78.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 74.0%

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

   4. Simplified 78.7%

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

      [Start] 74.0	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 74.0	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      associate--l+ [=>] 78.7	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   5. Taylor expanded in z around inf 75.5%

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

   6. Simplified 75.5%

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

      [Start] 75.5	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
      +-commutative [=>] 75.5	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Applied egg-rr 77.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right)} \]
      Proof

      [Start] 75.5	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) \]
      associate--r+ [=>] 77.8	\[ \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]
      expm1-log1p-u [=>] 77.8	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x}\right)\right)}\right) \]
      expm1-udef [=>] 77.8	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x}\right)} - 1\right)}\right) \]
      associate--r- [=>] 77.8	\[ \sqrt{x + 1} + \color{blue}{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right)} \]

   8. Applied egg-rr 85.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\color{blue}{\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      Proof

      [Start] 77.8	\[ \sqrt{x + 1} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      flip-- [=>] 79.8	\[ \sqrt{x + 1} + \left(\left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      div-inv [=>] 79.8	\[ \sqrt{x + 1} + \left(\left(\color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      add-sqr-sqrt [<=] 69.8	\[ \sqrt{x + 1} + \left(\left(\left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      +-commutative [=>] 69.8	\[ \sqrt{x + 1} + \left(\left(\left(\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      add-sqr-sqrt [<=] 85.7	\[ \sqrt{x + 1} + \left(\left(\left(\left(y + 1\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      associate--l+ [=>] 85.7	\[ \sqrt{x + 1} + \left(\left(\color{blue}{\left(y + \left(1 - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]

   9. Simplified 94.4%

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

      [Start] 85.7	\[ \sqrt{x + 1} + \left(\left(\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      associate-*r/ [=>] 85.7	\[ \sqrt{x + 1} + \left(\left(\color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      *-rgt-identity [=>] 85.7	\[ \sqrt{x + 1} + \left(\left(\frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      associate-+r- [=>] 85.7	\[ \sqrt{x + 1} + \left(\left(\frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      +-commutative [<=] 85.7	\[ \sqrt{x + 1} + \left(\left(\frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]
      associate--l+ [=>] 94.4	\[ \sqrt{x + 1} + \left(\left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right) + 1\right) \]

   if 1.00000000000000005e32 < y

   1. Initial program 79.2%

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

   2. Simplified 5.9%

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

      [Start] 79.2	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 79.2	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 79.2	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 79.2	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 79.2	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 79.2	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 79.2	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 79.2	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 74.5%

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

   4. Simplified 79.2%

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

      [Start] 74.5	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{z} + \sqrt{x}\right) - \sqrt{1 + z}\right)\right) \]
      +-commutative [=>] 74.5	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{z}\right)} - \sqrt{1 + z}\right)\right) \]
      associate--l+ [=>] 79.2	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)}\right) \]

   5. Taylor expanded in z around inf 74.4%

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

   6. Simplified 74.4%

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

      [Start] 74.4	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
      +-commutative [=>] 74.4	\[ \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Taylor expanded in y around inf 79.2%

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

   8. Applied egg-rr 96.8%

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

      [Start] 79.2	\[ \sqrt{1 + x} - \sqrt{x} \]
      flip-- [=>] 79.6	\[ \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]
      div-inv [=>] 79.6	\[ \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
      add-sqr-sqrt [<=] 80.0	\[ \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      add-sqr-sqrt [<=] 80.6	\[ \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      associate--l+ [=>] 96.8	\[ \color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   9. Simplified 96.8%

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

      [Start] 96.8	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [=>] 96.8	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      +-inverses [=>] 96.8	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      metadata-eval [=>] 96.8	\[ \color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      *-lft-identity [=>] 96.8	\[ \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + 2\\ \mathbf{elif}\;y \leq 28000:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} - \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+32}:\\ \;\;\;\;\sqrt{x + 1} + \left(1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	66372

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

Alternative 2
Accuracy	98.7%
Cost	40136

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{1}{t_3 + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;t_1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_2} + \left(\left(t_3 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	40136

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.7 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + 2\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;t_1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} + \left(\left(t_2 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 4
Accuracy	98.0%
Cost	40004

\[\begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + t_1\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 10^{+43}:\\ \;\;\;\;t_2 + \left(1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_1} - e^{\mathsf{log1p}\left(\sqrt{x}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}}\\ \end{array} \]

Alternative 6
Accuracy	97.0%
Cost	27080

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.85 \cdot 10^{-31}:\\ \;\;\;\;2 + \left(t_1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} + \left(t_1 - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 7
Accuracy	98.0%
Cost	27080

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + 2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;1 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} + \left(t_1 - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 8
Accuracy	92.7%
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;z \leq 3.2 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + t_1\right) - \sqrt{x}\\ \end{array} \]

Alternative 9
Accuracy	96.2%
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{-86}:\\ \;\;\;\;2 + \left(t_1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\frac{1}{t_1 + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 10
Accuracy	91.5%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.2 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(t_1 - \left(\left(\sqrt{y} + \sqrt{z}\right) - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(t_1 - \sqrt{y}\right)\right) - \sqrt{x}\\ \end{array} \]

Alternative 11
Accuracy	91.5%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;z \leq 3.2 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 10^{+25}:\\ \;\;\;\;1 + \left(t_1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + t_1\right) - \sqrt{x}\\ \end{array} \]

Alternative 12
Accuracy	89.7%
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4350000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	90.9%
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 3100000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) - \sqrt{x}\\ \end{array} \]

Alternative 14
Accuracy	89.5%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2100000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(1 + x \cdot 0.5\right)\\ \end{array} \]

Alternative 15
Accuracy	89.4%
Cost	20040

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 780000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	89.4%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 110000000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 17
Accuracy	89.5%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 3900000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18
Accuracy	84.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 4100000000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19
Accuracy	64.2%
Cost	13248

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

Alternative 20
Accuracy	35.6%
Cost	13120

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

Alternative 21
Accuracy	35.0%
Cost	6848

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

Alternative 22
Accuracy	34.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023133 
(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))))