Main:z from

Average Accuracy: 91.8% → 99.3%

Time: 30.9s

Precision: binary64

Cost: 79232

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

Math

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

↓

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

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)) (sqrt y))) (t_2 (sqrt (+ x 1.0))))
   (-
    (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (- (sqrt (+ 1.0 t)) (sqrt t)))
    (/ (+ t_1 (+ (sqrt x) t_2)) (* (- (- (sqrt x)) t_2) t_1)))))

C

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

↓

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

Fortran

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

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = sqrt((1.0d0 + y)) + sqrt(y)
    t_2 = sqrt((x + 1.0d0))
    code = ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))) - ((t_1 + (sqrt(x) + t_2)) / ((-sqrt(x) - t_2) * t_1))
end function

Java

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

↓

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

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)) + math.sqrt(y)
	t_2 = math.sqrt((x + 1.0))
	return ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t))) - ((t_1 + (math.sqrt(x) + t_2)) / ((-math.sqrt(x) - t_2) * t_1))

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 = Float64(sqrt(Float64(1.0 + y)) + sqrt(y))
	t_2 = sqrt(Float64(x + 1.0))
	return Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) - Float64(Float64(t_1 + Float64(sqrt(x) + t_2)) / Float64(Float64(Float64(-sqrt(x)) - t_2) * t_1)))
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 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y)) + sqrt(y);
	t_2 = sqrt((x + 1.0));
	tmp = ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t))) - ((t_1 + (sqrt(x) + t_2)) / ((-sqrt(x) - t_2) * t_1));
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[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 + N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Sqrt[x], $MachinePrecision]) - t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	91.8%
Target	99.4%
Herbie	99.3%

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

Derivation

1. Initial program 91.8%

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

2. Simplified 91.8%

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

   [Start] 91.8	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 91.8	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
   associate-+l- [=>] 91.8	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate--r- [=>] 90.6	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [<=] 90.6	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate-+l- [=>] 91.8	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [=>] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [=>] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [<=] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

3. Applied egg-rr 93.2%

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

   [Start] 91.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 92.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   div-inv [=>] 92.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 65.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 92.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

4. Simplified 93.2%

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

   [Start] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-lft-identity [=>] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

5. Applied egg-rr 94.4%

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

   [Start] 93.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 93.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   div-inv [=>] 93.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 83.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 93.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(1 + y\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(1 + \left(y - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Simplified 94.4%

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

   [Start] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-lft-identity [=>] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

7. Applied egg-rr 94.6%

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

   [Start] 94.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-exp-log [=>] 94.4	\[ \color{blue}{e^{\log \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--r- [=>] 94.6	\[ e^{\log \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 94.6	\[ e^{\log \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

8. Applied egg-rr 94.9%

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

   [Start] 94.6	\[ e^{\log \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-exp-log [<=] 94.6	\[ \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   frac-2neg [=>] 94.6	\[ \left(\color{blue}{\frac{-1}{-\left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 94.6	\[ \left(\frac{-1}{-\left(\sqrt{1 + y} + \sqrt{y}\right)} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   frac-add [=>] 94.6	\[ \color{blue}{\frac{\left(-1\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

9. Simplified 99.3%

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

   [Start] 94.9	\[ \frac{-1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right) + \left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   cancel-sign-sub-inv [<=] 94.9	\[ \frac{\color{blue}{-1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x + \left(1 - x\right)\right)}}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   mul-1-neg [=>] 94.9	\[ \frac{\color{blue}{\left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right)} - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 94.9	\[ \frac{\left(-\color{blue}{\left(\sqrt{x} + \sqrt{1 + x}\right)}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   distribute-neg-in [=>] 94.9	\[ \frac{\color{blue}{\left(\left(-\sqrt{x}\right) + \left(-\sqrt{1 + x}\right)\right)} - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   unsub-neg [=>] 94.9	\[ \frac{\color{blue}{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right)} - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 94.9	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \color{blue}{\left(\left(1 - x\right) + x\right)}}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+l- [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \color{blue}{\left(1 - \left(x - x\right)\right)}}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(1 - \color{blue}{0}\right)}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \color{blue}{1}}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-rgt-identity [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   distribute-lft-neg-out [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right)}{\color{blue}{-\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   distribute-rgt-neg-in [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right)}{\color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right)}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\left(\left(-\sqrt{x}\right) - \sqrt{1 + x}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(-\color{blue}{\left(\sqrt{x} + \sqrt{1 + x}\right)}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

10. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	92484

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;\left(\left(t_3 - \sqrt{x}\right) - \left(\sqrt{y} - t_1\right)\right) + \left(t_2 - \sqrt{z}\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(t_3 + \left(\frac{1}{t_1 + \sqrt{y}} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	79168

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

Alternative 3
Accuracy	97.6%
Cost	66116

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

Alternative 4
Accuracy	97.8%
Cost	40004

\[\begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+32}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 5
Accuracy	96.0%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 6
Accuracy	96.7%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 7
Accuracy	95.5%
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 8
Accuracy	93.1%
Cost	26820

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\left(t_1 + 2\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{t_1 + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \]

Alternative 9
Accuracy	91.1%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;1 + \left(t_1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	91.2%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 9.5 \cdot 10^{-32}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 11
Accuracy	90.4%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 8 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(1 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\left(t_1 - {z}^{0.5}\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	91.8%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 3100000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 13
Accuracy	90.2%
Cost	26628

\[\begin{array}{l} \mathbf{if}\;t \leq 4050000000:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\left(\sqrt{1 + z} - {z}^{0.5}\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	91.0%
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 1900000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 15
Accuracy	89.4%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;1 + \left(1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+26}:\\ \;\;\;\;1 - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 16
Accuracy	89.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 3.9 \cdot 10^{-31}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 3400000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \end{array} \]

Alternative 17
Accuracy	62.3%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;y \leq 6.6:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Accuracy	81.2%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 0.175:\\ \;\;\;\;\left(t_1 + 2\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{y} - t_1\right)\\ \end{array} \]

Alternative 19
Accuracy	84.9%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 1850000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \end{array} \]

Alternative 20
Accuracy	65.2%
Cost	13248

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

Alternative 21
Accuracy	35.8%
Cost	13120

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

Alternative 22
Accuracy	35.2%
Cost	6848

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

Alternative 23
Accuracy	34.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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))))