Main:z from

Average Error: 5.1 → 2.2

Time: 28.3s

Precision: binary64

Cost: 105476

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{y} - t_3\\ t_5 := \sqrt{1 + z}\\ \mathbf{if}\;\left(t_2 - \sqrt{x}\right) + \left(t_3 - \sqrt{y}\right) \leq 0.999996:\\ \;\;\;\;\frac{1 + \left(x - {\left(\sqrt{x} + t_4\right)}^{2}\right)}{\sqrt{x} + \left(t_2 + t_4\right)} + \left(\left(t_5 - \sqrt{z}\right) + \left(t_1 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_3} - \sqrt{x}\right)\right) + \left(\frac{1}{t_5 + \sqrt{z}} + \frac{1}{t_1 + \sqrt{t}}\right)\\ \end{array} \]

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 t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (- (sqrt y) t_3))
        (t_5 (sqrt (+ 1.0 z))))
   (if (<= (+ (- t_2 (sqrt x)) (- t_3 (sqrt y))) 0.999996)
     (+
      (/ (+ 1.0 (- x (pow (+ (sqrt x) t_4) 2.0))) (+ (sqrt x) (+ t_2 t_4)))
      (+ (- t_5 (sqrt z)) (- t_1 (sqrt t))))
     (+
      (+ t_2 (- (/ (+ 1.0 (- y y)) (+ (sqrt y) t_3)) (sqrt x)))
      (+ (/ 1.0 (+ t_5 (sqrt z))) (/ 1.0 (+ t_1 (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 + t));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt(y) - t_3;
	double t_5 = sqrt((1.0 + z));
	double tmp;
	if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 0.999996) {
		tmp = ((1.0 + (x - pow((sqrt(x) + t_4), 2.0))) / (sqrt(x) + (t_2 + t_4))) + ((t_5 - sqrt(z)) + (t_1 - sqrt(t)));
	} else {
		tmp = (t_2 + (((1.0 + (y - y)) / (sqrt(y) + t_3)) - sqrt(x))) + ((1.0 / (t_5 + sqrt(z))) + (1.0 / (t_1 + sqrt(t))));
	}
	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) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + y))
    t_4 = sqrt(y) - t_3
    t_5 = sqrt((1.0d0 + z))
    if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 0.999996d0) then
        tmp = ((1.0d0 + (x - ((sqrt(x) + t_4) ** 2.0d0))) / (sqrt(x) + (t_2 + t_4))) + ((t_5 - sqrt(z)) + (t_1 - sqrt(t)))
    else
        tmp = (t_2 + (((1.0d0 + (y - y)) / (sqrt(y) + t_3)) - sqrt(x))) + ((1.0d0 / (t_5 + sqrt(z))) + (1.0d0 / (t_1 + sqrt(t))))
    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 + t));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + y));
	double t_4 = Math.sqrt(y) - t_3;
	double t_5 = Math.sqrt((1.0 + z));
	double tmp;
	if (((t_2 - Math.sqrt(x)) + (t_3 - Math.sqrt(y))) <= 0.999996) {
		tmp = ((1.0 + (x - Math.pow((Math.sqrt(x) + t_4), 2.0))) / (Math.sqrt(x) + (t_2 + t_4))) + ((t_5 - Math.sqrt(z)) + (t_1 - Math.sqrt(t)));
	} else {
		tmp = (t_2 + (((1.0 + (y - y)) / (Math.sqrt(y) + t_3)) - Math.sqrt(x))) + ((1.0 / (t_5 + Math.sqrt(z))) + (1.0 / (t_1 + Math.sqrt(t))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

↓

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + y))
	t_4 = math.sqrt(y) - t_3
	t_5 = math.sqrt((1.0 + z))
	tmp = 0
	if ((t_2 - math.sqrt(x)) + (t_3 - math.sqrt(y))) <= 0.999996:
		tmp = ((1.0 + (x - math.pow((math.sqrt(x) + t_4), 2.0))) / (math.sqrt(x) + (t_2 + t_4))) + ((t_5 - math.sqrt(z)) + (t_1 - math.sqrt(t)))
	else:
		tmp = (t_2 + (((1.0 + (y - y)) / (math.sqrt(y) + t_3)) - math.sqrt(x))) + ((1.0 / (t_5 + math.sqrt(z))) + (1.0 / (t_1 + 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 + t))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(sqrt(y) - t_3)
	t_5 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(x)) + Float64(t_3 - sqrt(y))) <= 0.999996)
		tmp = Float64(Float64(Float64(1.0 + Float64(x - (Float64(sqrt(x) + t_4) ^ 2.0))) / Float64(sqrt(x) + Float64(t_2 + t_4))) + Float64(Float64(t_5 - sqrt(z)) + Float64(t_1 - sqrt(t))));
	else
		tmp = Float64(Float64(t_2 + Float64(Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(y) + t_3)) - sqrt(x))) + Float64(Float64(1.0 / Float64(t_5 + sqrt(z))) + Float64(1.0 / Float64(t_1 + sqrt(t)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + y));
	t_4 = sqrt(y) - t_3;
	t_5 = sqrt((1.0 + z));
	tmp = 0.0;
	if (((t_2 - sqrt(x)) + (t_3 - sqrt(y))) <= 0.999996)
		tmp = ((1.0 + (x - ((sqrt(x) + t_4) ^ 2.0))) / (sqrt(x) + (t_2 + t_4))) + ((t_5 - sqrt(z)) + (t_1 - sqrt(t)));
	else
		tmp = (t_2 + (((1.0 + (y - y)) / (sqrt(y) + t_3)) - sqrt(x))) + ((1.0 / (t_5 + sqrt(z))) + (1.0 / (t_1 + sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.999996], N[(N[(N[(1.0 + N[(x - N[Power[N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$5 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	5.1
Target	0.3
Herbie	2.2

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

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))) < 0.999995999999999996

   1. Initial program 47.9

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

   2. Simplified 47.9

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

      [Start] 47.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 47.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 47.9	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 56.6	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 56.6	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 47.9	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 47.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 47.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 47.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 47.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 47.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 56.6

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

   4. Simplified 56.6

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

      [Start] 56.6	\[ \left(\sqrt{1 + x} + \left(-\left(\sqrt{y} + \left(\sqrt{x} - \sqrt{1 + y}\right)\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      unsub-neg [=>] 56.6	\[ \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{y} + \left(\sqrt{x} - \sqrt{1 + y}\right)\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 46.3

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - {\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}^{2}\right) \cdot \frac{1}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Simplified 30.8

      \[\leadsto \color{blue}{\frac{1 + \left(x - {\left(\sqrt{x} + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)}^{2}\right)}{\sqrt{x} + \left(\left(\sqrt{y} - \sqrt{1 + y}\right) + \sqrt{x + 1}\right)}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      Proof

      [Start] 46.3	\[ \left(\left(x + 1\right) - {\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}^{2}\right) \cdot \frac{1}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*r/ [=>] 46.3	\[ \color{blue}{\frac{\left(\left(x + 1\right) - {\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}^{2}\right) \cdot 1}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-rgt-identity [=>] 46.3	\[ \frac{\color{blue}{\left(x + 1\right) - {\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}^{2}}}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      unpow2 [=>] 46.3	\[ \frac{\left(x + 1\right) - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) \cdot \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      sqr-neg [<=] 46.3	\[ \frac{\left(x + 1\right) - \color{blue}{\left(-\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right) \cdot \left(-\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)}}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \sqrt{x + 1}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   if 0.999995999999999996 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)))

   1. Initial program 2.0

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

   2. Simplified 2.0

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

      [Start] 2.0	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.0	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 2.0	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 2.2	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 2.2	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 2.0	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 2.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 2.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 2.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 1.6

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

   4. Simplified 1.2

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

      [Start] 1.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*r/ [=>] 1.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-rgt-identity [=>] 1.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 1.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [<=] 1.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 0.3

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

   6. Simplified 0.3

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

      [Start] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Applied egg-rr 0.0

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

   8. Simplified 0.0

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

      [Start] 0.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 0.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\left(t - t\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 0.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 0.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.999996:\\ \;\;\;\;\frac{1 + \left(x - {\left(\sqrt{x} + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)}^{2}\right)}{\sqrt{x} + \left(\sqrt{x + 1} + \left(\sqrt{y} - \sqrt{1 + y}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.8
Cost	65988

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

Alternative 2
Error	3.2
Cost	53312

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

Alternative 3
Error	3.4
Cost	53184

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

Alternative 4
Error	4.4
Cost	53056

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

Alternative 5
Error	4.2
Cost	52800

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

Alternative 6
Error	3.8
Cost	40004

\[\begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 10^{+84}:\\ \;\;\;\;\left(t_2 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(t_1 + \left(t_2 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	4.1
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 10^{+16}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{t_1 + \sqrt{t}}\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y} + t_2} + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 8
Error	3.7
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;\left(t_2 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(t_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y} + t_1} + \left(t_2 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 9
Error	4.7
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 6600000000000:\\ \;\;\;\;t_1 + \left(\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 10
Error	4.4
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_2\right) + \left(1 + \left(t_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{\sqrt{y} + t_1} + \left(t_2 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 11
Error	5.6
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;z \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\left(2 + \left(1 + z \cdot 0.5\right)\right) - \left(\left(\sqrt{z} + \sqrt{t}\right) - \sqrt{1 + t}\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+89}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + t_1\\ \end{array} \]

Alternative 12
Error	5.4
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1000000000000:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + t_1\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(t_1 - \sqrt{z}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \end{array} \]

Alternative 13
Error	6.5
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;\left(2 + \left(1 + z \cdot 0.5\right)\right) - \left(\left(\sqrt{z} + \sqrt{t}\right) - \sqrt{1 + t}\right)\\ \mathbf{elif}\;z \leq 5200000000000:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 14
Error	5.7
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{-21}:\\ \;\;\;\;\left(2 + \left(1 + z \cdot 0.5\right)\right) - \left(\left(\sqrt{z} + \sqrt{t}\right) - \sqrt{1 + t}\right)\\ \mathbf{elif}\;z \leq 8200000000000:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15
Error	5.4
Cost	26564

\[\begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 16
Error	6.6
Cost	20292

\[\begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;\left(2 + \left(1 + z \cdot 0.5\right)\right) - \left(\left(\sqrt{z} + \sqrt{t}\right) - \sqrt{1 + t}\right)\\ \mathbf{elif}\;z \leq 31000000000000:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 17
Error	6.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 5500000000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18
Error	6.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 12600000000000:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19
Error	9.9
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 21000000000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 20
Error	23.1
Cost	13248

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

Alternative 21
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023016 
(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))))