Main:z from

Percentage Accurate: 91.8% → 99.4%

Time: 33.0s

Precision: binary64

Cost: 92228

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{x} + \sqrt{1 + x}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;t_2 - \sqrt{y} \leq 0.2:\\ \;\;\;\;\frac{\frac{\sqrt{y} + \left(t_1 + t_2\right)}{\sqrt{y} + t_2}}{t_1} + \left(\left(t_3 - \sqrt{z}\right) + \left(t_4 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t_2\right) - \sqrt{y}\right) + \left(\frac{1}{t_3 + \sqrt{z}} + \frac{1}{t_4 + \sqrt{t}}\right)\\ \end{array} \]

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 x) (sqrt (+ 1.0 x))))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= (- t_2 (sqrt y)) 0.2)
     (+
      (/ (/ (+ (sqrt y) (+ t_1 t_2)) (+ (sqrt y) t_2)) t_1)
      (+ (- t_3 (sqrt z)) (- t_4 (sqrt t))))
     (+
      (- (+ 1.0 t_2) (sqrt y))
      (+ (/ 1.0 (+ t_3 (sqrt z))) (/ 1.0 (+ t_4 (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(x) + sqrt((1.0 + x));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if ((t_2 - sqrt(y)) <= 0.2) {
		tmp = (((sqrt(y) + (t_1 + t_2)) / (sqrt(y) + t_2)) / t_1) + ((t_3 - sqrt(z)) + (t_4 - sqrt(t)));
	} else {
		tmp = ((1.0 + t_2) - sqrt(y)) + ((1.0 / (t_3 + sqrt(z))) + (1.0 / (t_4 + 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) :: tmp
    t_1 = sqrt(x) + sqrt((1.0d0 + x))
    t_2 = sqrt((y + 1.0d0))
    t_3 = sqrt((1.0d0 + z))
    t_4 = sqrt((1.0d0 + t))
    if ((t_2 - sqrt(y)) <= 0.2d0) then
        tmp = (((sqrt(y) + (t_1 + t_2)) / (sqrt(y) + t_2)) / t_1) + ((t_3 - sqrt(z)) + (t_4 - sqrt(t)))
    else
        tmp = ((1.0d0 + t_2) - sqrt(y)) + ((1.0d0 / (t_3 + sqrt(z))) + (1.0d0 / (t_4 + 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(x) + Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if ((t_2 - Math.sqrt(y)) <= 0.2) {
		tmp = (((Math.sqrt(y) + (t_1 + t_2)) / (Math.sqrt(y) + t_2)) / t_1) + ((t_3 - Math.sqrt(z)) + (t_4 - Math.sqrt(t)));
	} else {
		tmp = ((1.0 + t_2) - Math.sqrt(y)) + ((1.0 / (t_3 + Math.sqrt(z))) + (1.0 / (t_4 + 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(x) + math.sqrt((1.0 + x))
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((1.0 + z))
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if (t_2 - math.sqrt(y)) <= 0.2:
		tmp = (((math.sqrt(y) + (t_1 + t_2)) / (math.sqrt(y) + t_2)) / t_1) + ((t_3 - math.sqrt(z)) + (t_4 - math.sqrt(t)))
	else:
		tmp = ((1.0 + t_2) - math.sqrt(y)) + ((1.0 / (t_3 + math.sqrt(z))) + (1.0 / (t_4 + 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 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(y)) <= 0.2)
		tmp = Float64(Float64(Float64(Float64(sqrt(y) + Float64(t_1 + t_2)) / Float64(sqrt(y) + t_2)) / t_1) + Float64(Float64(t_3 - sqrt(z)) + Float64(t_4 - sqrt(t))));
	else
		tmp = Float64(Float64(Float64(1.0 + t_2) - sqrt(y)) + Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + Float64(1.0 / Float64(t_4 + 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(x) + sqrt((1.0 + x));
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((1.0 + z));
	t_4 = sqrt((1.0 + t));
	tmp = 0.0;
	if ((t_2 - sqrt(y)) <= 0.2)
		tmp = (((sqrt(y) + (t_1 + t_2)) / (sqrt(y) + t_2)) / t_1) + ((t_3 - sqrt(z)) + (t_4 - sqrt(t)));
	else
		tmp = ((1.0 + t_2) - sqrt(y)) + ((1.0 / (t_3 + sqrt(z))) + (1.0 / (t_4 + 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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	91.8%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.20000000000000001

   1. Initial program 86.9%

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

   2. Simplified 86.5%

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

      [Start] 86.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 86.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 86.5	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{\log \left(e^{\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      Step-by-step derivation

      [Start] 86.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-log-exp [=>] 86.5	\[ \color{blue}{\log \left(e^{\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 86.9	\[ \log \left(e^{\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 86.9	\[ \log \left(e^{\left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   4. Applied egg-rr 87.7%

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

      [Start] 86.9	\[ \log \left(e^{\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 86.8	\[ \log \left(e^{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 86.8	\[ \log \left(e^{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 72.9	\[ \log \left(e^{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 87.3	\[ \log \left(e^{\left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 87.7	\[ \log \left(e^{\color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Simplified 87.7%

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

      [Start] 87.7	\[ \log \left(e^{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 87.7	\[ \log \left(e^{\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 87.7	\[ \log \left(e^{\color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 87.7	\[ \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Applied egg-rr 91.7%

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

      [Start] 87.7	\[ \log \left(e^{\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-log-exp [<=] 89.2	\[ \color{blue}{\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 89.1	\[ \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      frac-add [=>] 89.1	\[ \color{blue}{\frac{1 \cdot \left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-un-lft-identity [<=] 89.1	\[ \frac{\color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 52.8	\[ \frac{\left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 89.3	\[ \frac{\left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - \color{blue}{y}\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 91.7	\[ \frac{\left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \color{blue}{\left(1 + \left(y - y\right)\right)}}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 91.7%

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

      [Start] 91.7	\[ \frac{\left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(1 + \left(y - y\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-/l/ [<=] 91.7	\[ \color{blue}{\frac{\frac{\left(\sqrt{1 + y} + \sqrt{y}\right) + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(1 + \left(y - y\right)\right)}{\sqrt{1 + y} + \sqrt{y}}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 91.7	\[ \frac{\frac{\color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(1 + \left(y - y\right)\right) + \left(\sqrt{1 + y} + \sqrt{y}\right)}}{\sqrt{1 + y} + \sqrt{y}}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-commutative [<=] 91.7	\[ \frac{\frac{\color{blue}{\left(1 + \left(y - y\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(\sqrt{1 + y} + \sqrt{y}\right)}{\sqrt{1 + y} + \sqrt{y}}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-rgt-identity [<=] 91.7	\[ \frac{\frac{\left(1 + \left(y - y\right)\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right) + \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot 1}}{\sqrt{1 + y} + \sqrt{y}}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   if 0.20000000000000001 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 96.9%

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

   2. Simplified 59.9%

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

      [Start] 96.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 96.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 59.9	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 60.3%

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

      [Start] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 48.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 60.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   4. Simplified 60.3%

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

      [Start] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 61.0%

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

      [Start] 60.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 60.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      div-inv [=>] 60.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      add-sqr-sqrt [<=] 40.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 60.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      associate--l+ [=>] 61.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 + \left(t - t\right)\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   6. Simplified 61.0%

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

      [Start] 61.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 61.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 61.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 61.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   7. Taylor expanded in x around 0 58.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0.2:\\ \;\;\;\;\frac{\frac{\sqrt{y} + \left(\left(\sqrt{x} + \sqrt{1 + x}\right) + \sqrt{y + 1}\right)}{\sqrt{y} + \sqrt{y + 1}}}{\sqrt{x} + \sqrt{1 + x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \sqrt{y + 1}\right) - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	91524

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;t_3 - \sqrt{x} \leq 0.02:\\ \;\;\;\;\left(\left(t_1 - \sqrt{z}\right) + \left(t_2 - \sqrt{t}\right)\right) + {\left(\mathsf{hypot}\left(\sqrt{t_4}, {\left(\sqrt{x} + t_3\right)}^{-0.5}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{z}} + \frac{1}{t_2 + \sqrt{t}}\right) + \left(t_3 + \left(t_4 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	97.5%
Cost	66244

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

Alternative 3
Accuracy	97.7%
Cost	66116

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.02:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + \left(t_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	98.0%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 7 \cdot 10^{-36}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t_1 + \sqrt{z}}\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+32}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 5
Accuracy	96.9%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(1 + \sqrt{y + 1}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	39752

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-36}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{t_1 + \sqrt{z}}\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 7
Accuracy	95.7%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{elif}\;y \leq 10^{+28}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(t_1 - \sqrt{x}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 8
Accuracy	96.1%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 10^{-25}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{elif}\;y \leq 10^{+28}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(t_1 - \sqrt{x}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 9
Accuracy	90.0%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+28}:\\ \;\;\;\;t_1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \sqrt{y + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 10
Accuracy	90.0%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+27}:\\ \;\;\;\;\left(\sqrt{y + 1} + \left(t_1 - \sqrt{x}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 11
Accuracy	64.6%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-216}:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;2 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 12
Accuracy	82.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 0.108:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 13
Accuracy	84.5%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14
Accuracy	86.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 15
Accuracy	63.4%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-217}:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;2 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	62.0%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;2 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	34.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023161 
(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))))