Main:z from

Average Error: 8.54% → 0.41%

Time: 30.4s

Precision: binary64

Cost: 92548

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{y} - t_1\\ t_5 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \mathbf{if}\;\left(t_3 - \sqrt{x}\right) - t_4 \leq 1.2:\\ \;\;\;\;{\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{y} + t_1} + \frac{1}{\sqrt{x} + t_3}}\right)}^{3} + \left(t_5 + \left(t_2 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_3 - \left(\sqrt{x} + t_4\right)\right) + \left(t_5 + \frac{1}{t_2 + \sqrt{t}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 t)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (- (sqrt y) t_1))
        (t_5 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))))
   (if (<= (- (- t_3 (sqrt x)) t_4) 1.2)
     (+
      (pow
       (cbrt (+ (/ (+ 1.0 (- y y)) (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt x) t_3))))
       3.0)
      (+ t_5 (- t_2 (sqrt t))))
     (+ (- t_3 (+ (sqrt x) t_4)) (+ t_5 (/ 1.0 (+ t_2 (sqrt t))))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t));
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt(y) - t_1;
	double t_5 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double tmp;
	if (((t_3 - sqrt(x)) - t_4) <= 1.2) {
		tmp = pow(cbrt((((1.0 + (y - y)) / (sqrt(y) + t_1)) + (1.0 / (sqrt(x) + t_3)))), 3.0) + (t_5 + (t_2 - sqrt(t)));
	} else {
		tmp = (t_3 - (sqrt(x) + t_4)) + (t_5 + (1.0 / (t_2 + sqrt(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 + y));
	double t_2 = Math.sqrt((1.0 + t));
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = Math.sqrt(y) - t_1;
	double t_5 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double tmp;
	if (((t_3 - Math.sqrt(x)) - t_4) <= 1.2) {
		tmp = Math.pow(Math.cbrt((((1.0 + (y - y)) / (Math.sqrt(y) + t_1)) + (1.0 / (Math.sqrt(x) + t_3)))), 3.0) + (t_5 + (t_2 - Math.sqrt(t)));
	} else {
		tmp = (t_3 - (Math.sqrt(x) + t_4)) + (t_5 + (1.0 / (t_2 + Math.sqrt(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 + y))
	t_2 = sqrt(Float64(1.0 + t))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(sqrt(y) - t_1)
	t_5 = Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	tmp = 0.0
	if (Float64(Float64(t_3 - sqrt(x)) - t_4) <= 1.2)
		tmp = Float64((cbrt(Float64(Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(x) + t_3)))) ^ 3.0) + Float64(t_5 + Float64(t_2 - sqrt(t))));
	else
		tmp = Float64(Float64(t_3 - Float64(sqrt(x) + t_4)) + Float64(t_5 + Float64(1.0 / Float64(t_2 + sqrt(t)))));
	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[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision], 1.2], N[(N[Power[N[Power[N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(t$95$5 + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + N[(1.0 / N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	8.54%
Target	0.63%
Herbie	0.41%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 1.19999999999999996

   1. Initial program 21.64

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

   2. Simplified 21.64

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

      [Start] 21.64	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 21.64	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 21.64	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 25.56	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 25.56	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 21.64	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 21.64	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 21.64	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 21.64	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 21.64	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 21.64	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 21.74

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

   4. Applied egg-rr 7.36

      \[\leadsto {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Simplified 7.36

      \[\leadsto {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      Proof

      [Start] 7.36	\[ {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 7.36	\[ {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 7.36	\[ {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 7.36	\[ {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 7.36	\[ {\left(\sqrt[3]{\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Applied egg-rr 5.72

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 1.78

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      Proof

      [Start] 5.72	\[ {\left(\sqrt[3]{\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*r/ [=>] 5.72	\[ {\left(\sqrt[3]{\color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-rgt-identity [=>] 5.72	\[ {\left(\sqrt[3]{\frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 5.73	\[ {\left(\sqrt[3]{\frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [<=] 5.73	\[ {\left(\sqrt[3]{\frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 1.78	\[ {\left(\sqrt[3]{\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Applied egg-rr 1.06

      \[\leadsto {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\color{blue}{\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   9. Simplified 1.06

      \[\leadsto {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      Proof

      [Start] 1.06	\[ {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 1.06	\[ {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 1.06	\[ {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 1.06	\[ {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 1.06	\[ {\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   if 1.19999999999999996 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)))

   1. Initial program 2.48

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

   2. Simplified 2.48

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

      [Start] 2.48	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.48	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 2.48	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 2.48	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 2.48	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 2.48	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.48	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 2.48	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 2.48	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 2.48	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.48	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 0.81

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

   4. Simplified 0.81

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

      [Start] 0.81	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.81	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.81	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.81	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 0.81	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 0.11

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

   6. Simplified 0.11

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

      [Start] 0.11	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 0.11	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\left(t - t\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 0.11	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 0.11	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 0.11	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.41

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{1 + y}\right) \leq 1.2:\\ \;\;\;\;{\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{x + 1}}}\right)}^{3} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \left(\sqrt{x} + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.64%
Cost	92420

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{y} - t_3\\ t_5 := \sqrt{1 + t}\\ \mathbf{if}\;\left(t_2 - \sqrt{x}\right) - t_4 \leq 1.2:\\ \;\;\;\;{\left(\sqrt[3]{\frac{1 + \left(y - y\right)}{\sqrt{y} + t_3} + \frac{1}{\sqrt{x} + t_2}}\right)}^{3} + \left(\left(t_5 - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 - \left(\sqrt{x} + t_4\right)\right) + \left(\frac{1}{t_1 + \sqrt{z}} + \frac{1}{t_5 + \sqrt{t}}\right)\\ \end{array} \]

Alternative 2
Error	4.7%
Cost	78980

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x + 1}\\ t_3 := \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 1:\\ \;\;\;\;t_3 + {\left(\sqrt[3]{\frac{1}{\sqrt{x} + t_2} - \left(\sqrt{y} - t_1\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(1 + \frac{1}{\sqrt{y} + t_1}\right)\\ \end{array} \]

Alternative 3
Error	5.74%
Cost	78852

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_3 - \sqrt{x} \leq 1:\\ \;\;\;\;\left(t_4 + \left(t_2 - \sqrt{z}\right)\right) + {\left(\sqrt[3]{\frac{1}{\sqrt{x} + t_3} - \left(\sqrt{y} - t_1\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{z}} + t_4\right) + \left(1 + \frac{1}{\sqrt{y} + t_1}\right)\\ \end{array} \]

Alternative 4
Error	2.09%
Cost	53060

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.9999999999999:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]

Alternative 5
Error	3.23%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 - \left(\sqrt{y} - \sqrt{1 + y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 6
Error	3.13%
Cost	33476

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(2 + y \cdot 0.5\right) - \sqrt{y}\right) + \left(\frac{1}{t_1 + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + \left(t_1 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 7
Error	2.45%
Cost	33476

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 10^{-6}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\right) + \left(\left(2 + y \cdot 0.5\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 8
Error	3.57%
Cost	33348

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\left(t_1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(2 + y \cdot 0.5\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 9
Error	4.09%
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(t_1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(1 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 10
Error	9.76%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;2 + \left(\left(\mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right) - \sqrt{z}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 11
Error	9.76%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 7.6 \cdot 10^{-12}:\\ \;\;\;\;2 + \left(\left(\mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right) - \sqrt{z}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} - \left(\sqrt{x} - \left(t_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 12
Error	4.76%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} - \left(\sqrt{x} - \left(t_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 13
Error	9.92%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-10}:\\ \;\;\;\;2 + \left(\left(\mathsf{fma}\left(0.5, y, \sqrt{1 + z}\right) - \sqrt{z}\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 14
Error	22.78%
Cost	20428

\[\begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(2 + y \cdot 0.5\right) - \sqrt{y}\right) + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left(x \cdot 0.5 + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 15
Error	27.44%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left(x \cdot 0.5 + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 16
Error	27.53%
Cost	20164

\[\begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + y \cdot 0.5\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 17
Error	29.69%
Cost	19908

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

Alternative 18
Error	38.03%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.65:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	34.41%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 20
Error	29.79%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 21
Error	64.07%
Cost	13120

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

Alternative 22
Error	65.23%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(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))))