Main:z from

Average Error: 5.6 → 0.5

Time: 32.2s

Precision: binary64

Cost: 67012

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 3.6634638227101813 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\left(t_4 - \sqrt{x}\right) + \left(t_1 - \sqrt{y}\right)\right) + \frac{1}{t_2 + \sqrt{z}}\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\left(\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \left(x + 1\right)\right) - \sqrt{x} \cdot t_4\right) + \frac{1 + \left(y - y\right)}{\sqrt{y} + t_1}\right) + \left(t_2 - \sqrt{z}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ x 1.0))))
   (if (<= z 3.6634638227101813e+28)
     (+ (+ (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))) (/ 1.0 (+ t_2 (sqrt z)))) t_3)
     (+
      t_3
      (+
       (+
        (*
         (/ 1.0 (+ (pow (+ x 1.0) 1.5) (pow x 1.5)))
         (- (+ x (+ x 1.0)) (* (sqrt x) t_4)))
        (/ (+ 1.0 (- y y)) (+ (sqrt y) t_1)))
       (- t_2 (sqrt z)))))))

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((z + 1.0));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((x + 1.0));
	double tmp;
	if (z <= 3.6634638227101813e+28) {
		tmp = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + (1.0 / (t_2 + sqrt(z)))) + t_3;
	} else {
		tmp = t_3 + ((((1.0 / (pow((x + 1.0), 1.5) + pow(x, 1.5))) * ((x + (x + 1.0)) - (sqrt(x) * t_4))) + ((1.0 + (y - y)) / (sqrt(y) + t_1))) + (t_2 - sqrt(z)));
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = sqrt((x + 1.0d0))
    if (z <= 3.6634638227101813d+28) then
        tmp = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + (1.0d0 / (t_2 + sqrt(z)))) + t_3
    else
        tmp = t_3 + ((((1.0d0 / (((x + 1.0d0) ** 1.5d0) + (x ** 1.5d0))) * ((x + (x + 1.0d0)) - (sqrt(x) * t_4))) + ((1.0d0 + (y - y)) / (sqrt(y) + t_1))) + (t_2 - sqrt(z)))
    end if
    code = tmp
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));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = Math.sqrt((x + 1.0));
	double tmp;
	if (z <= 3.6634638227101813e+28) {
		tmp = (((t_4 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + (1.0 / (t_2 + Math.sqrt(z)))) + t_3;
	} else {
		tmp = t_3 + ((((1.0 / (Math.pow((x + 1.0), 1.5) + Math.pow(x, 1.5))) * ((x + (x + 1.0)) - (Math.sqrt(x) * t_4))) + ((1.0 + (y - y)) / (Math.sqrt(y) + t_1))) + (t_2 - Math.sqrt(z)));
	}
	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 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = math.sqrt((x + 1.0))
	tmp = 0
	if z <= 3.6634638227101813e+28:
		tmp = (((t_4 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + (1.0 / (t_2 + math.sqrt(z)))) + t_3
	else:
		tmp = t_3 + ((((1.0 / (math.pow((x + 1.0), 1.5) + math.pow(x, 1.5))) * ((x + (x + 1.0)) - (math.sqrt(x) * t_4))) + ((1.0 + (y - y)) / (math.sqrt(y) + t_1))) + (t_2 - math.sqrt(z)))
	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(z + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (z <= 3.6634638227101813e+28)
		tmp = Float64(Float64(Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(1.0 / Float64(t_2 + sqrt(z)))) + t_3);
	else
		tmp = Float64(t_3 + Float64(Float64(Float64(Float64(1.0 / Float64((Float64(x + 1.0) ^ 1.5) + (x ^ 1.5))) * Float64(Float64(x + Float64(x + 1.0)) - Float64(sqrt(x) * t_4))) + Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(y) + t_1))) + Float64(t_2 - sqrt(z))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = sqrt((x + 1.0));
	tmp = 0.0;
	if (z <= 3.6634638227101813e+28)
		tmp = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + (1.0 / (t_2 + sqrt(z)))) + t_3;
	else
		tmp = t_3 + ((((1.0 / (((x + 1.0) ^ 1.5) + (x ^ 1.5))) * ((x + (x + 1.0)) - (sqrt(x) * t_4))) + ((1.0 + (y - y)) / (sqrt(y) + t_1))) + (t_2 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.6634638227101813e+28], N[(N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(t$95$3 + N[(N[(N[(N[(1.0 / N[(N[Power[N[(x + 1.0), $MachinePrecision], 1.5], $MachinePrecision] + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.6
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < 3.6634638227101813e28

   1. Initial program 3.0

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

   2. Applied egg-rr 2.1

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

   3. Taylor expanded in z around 0 1.0

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

   if 3.6634638227101813e28 < z

   1. Initial program 7.2

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

   2. Applied egg-rr 2.2

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

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.5

      \[\leadsto \left(\left(\color{blue}{\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)} + \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied egg-rr 0.2

      \[\leadsto \left(\left(\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(1 + x\right) + x\right) - \color{blue}{\sqrt{1 + x} \cdot \sqrt{x}}\right) + \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6634638227101813 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \left(x + 1\right)\right) - \sqrt{x} \cdot \sqrt{x + 1}\right) + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	79556

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

Alternative 2
Error	1.5
Cost	66628

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

Alternative 3
Error	0.5
Cost	53572

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 3.6634638227101813 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\left(t_4 - \sqrt{x}\right) + \left(t_1 - \sqrt{y}\right)\right) + \frac{1}{t_2 + \sqrt{z}}\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\left(t_2 - \sqrt{z}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_1} + \frac{1 + \left(x - x\right)}{\sqrt{x} + t_4}\right)\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	53188

\[\begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;t_1 \leq 10^{-7}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{z + \left(1 - z\right)}{\sqrt{z + 1} + \sqrt{z}} + \left(1 + t_1\right)\right)\\ \end{array} \]

Alternative 5
Error	1.8
Cost	53188

\[\begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 2522458.00101378:\\ \;\;\;\;\left(\left(\left(t_4 - \sqrt{x}\right) + t_1\right) + \frac{1}{t_2 + \sqrt{z}}\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\left(t_2 - \sqrt{z}\right) + \left(t_1 + \frac{1 + \left(x - x\right)}{\sqrt{x} + t_4}\right)\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 2522458.00101378:\\ \;\;\;\;\left(\left(\left(t_1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 7
Error	2.7
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 222920785924.61026:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 8
Error	6.8
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.0114918899052811 \cdot 10^{-22}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.088889138972062 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + t_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 9
Error	3.1
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.0114918899052811 \cdot 10^{-22}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\right)\\ \mathbf{elif}\;y \leq 4.088889138972062 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + t_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 10
Error	6.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;y \leq 1.0114918899052811 \cdot 10^{-22}:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 222920785924.61026:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternative 11
Error	12.1
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 0.12266591364158563:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 12
Error	10.4
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 378797530030433.6:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 13
Error	41.2
Cost	13120

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

Alternative 14
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(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))))