Main:z from

Average Accuracy: 91.7% → 98.9%

Time: 30.7s

Precision: binary64

Cost: 66372

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y} + \sqrt{y}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\left(\frac{1}{t_1} + \frac{1}{\sqrt{x} + t_2}\right) + \left(t_4 + \left(1 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\frac{1 + \left(y - y\right)}{t_1} + \left(\left(t_4 - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \end{array} \]

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

Target

Original	91.7%
Target	99.4%
Herbie	98.9%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.0

   1. Initial program 3.4%

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

   2. Simplified 3.4%

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

      [Start] 3.4	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 3.4	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 3.4	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 3.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 3.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 3.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 3.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 83.1%

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

      [Start] 3.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      flip-- [=>] 3.4	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      div-inv [=>] 3.4	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 4.2	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 4.2	\[ \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 3.8	\[ \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 83.1	\[ \color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 83.1	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [=>] 83.1	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      hypot-1-def [=>] 83.1	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   4. Simplified 83.1%

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

      [Start] 83.1	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 83.1	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 83.1	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 83.1	\[ \color{blue}{1} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 83.1	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 83.1	\[ \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   5. Applied egg-rr 83.1%

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

      [Start] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      flip-- [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      div-inv [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 41.2	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 41.2	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\left(y + 1\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\left(y + \left(1 - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   6. Simplified 98.2%

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

      [Start] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-*r/ [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-rgt-identity [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-+r- [=>] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [<=] 83.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 98.2	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   7. Taylor expanded in z around 0 3.4%

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

   8. Simplified 98.2%

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

      [Start] 3.4	\[ \left(1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \sqrt{1 + t}\right)\right)\right) - \sqrt{t} \]
      +-commutative [=>] 3.4	\[ \color{blue}{\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \sqrt{1 + t}\right)\right) + 1\right)} - \sqrt{t} \]
      associate--l+ [=>] 3.4	\[ \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \sqrt{1 + t}\right)\right) + \left(1 - \sqrt{t}\right)} \]
      associate-+r+ [=>] 3.4	\[ \color{blue}{\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) + \sqrt{1 + t}\right)} + \left(1 - \sqrt{t}\right) \]
      associate-+l+ [=>] 98.2	\[ \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) + \left(\sqrt{1 + t} + \left(1 - \sqrt{t}\right)\right)} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 96.4%

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

   2. Simplified 96.4%

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

      [Start] 96.4	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 96.4	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 96.4	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 96.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 96.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 96.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 96.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 96.9%

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

      [Start] 96.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      flip-- [=>] 96.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      div-inv [=>] 96.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 86.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 86.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(y + 1\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\left(y + \left(1 - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   4. Simplified 97.6%

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

      [Start] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-*r/ [=>] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-rgt-identity [=>] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-+r- [=>] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [<=] 96.9	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 97.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   5. Applied egg-rr 99.0%

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

      [Start] 97.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      flip-- [=>] 97.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      div-inv [=>] 97.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 69.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      add-sqr-sqrt [<=] 98.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   6. Simplified 99.0%

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

      [Start] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 99.0	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + t} + \left(1 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	79428

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

Alternative 2
Accuracy	98.0%
Cost	59520

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

Alternative 3
Accuracy	98.4%
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 5.65 \cdot 10^{+18}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \frac{-1}{t_3 + \sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{x} + t_2}\right) + \left(t_3 + \left(1 - \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(\left(t_1 - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(t_3 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_3 + \sqrt{y}} + \frac{1}{\sqrt{x} + t_2}\right) + \left(t_1 + \left(1 - \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	98.0%
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 5.65 \cdot 10^{+18}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\left(t_1 - \sqrt{t}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + t_2}\right) + \left(t_1 + \left(1 - \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	92.7%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{-27}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\frac{1}{t_1 + \sqrt{t}} + 2\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;t_2 + \left(\left(t_3 - \sqrt{y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1}{t_3 + \sqrt{y}} + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	96.2%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right)\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+30}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 8
Accuracy	96.7%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \frac{-1}{t_1 + \sqrt{t}}\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 9
Accuracy	97.2%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(t_1 + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+31}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 10
Accuracy	90.4%
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{-27}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	91.4%
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;z \leq 5.9 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 5.65 \cdot 10^{+18}:\\ \;\;\;\;t_1 + \left(1 + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 12
Accuracy	91.7%
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;z \leq 5.1 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + 2\right)\\ \mathbf{elif}\;z \leq 5.65 \cdot 10^{+18}:\\ \;\;\;\;t_1 + \left(1 + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 13
Accuracy	89.8%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+23}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 14
Accuracy	89.7%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + y} + t_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 15
Accuracy	90.8%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;z \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 1080000000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 16
Accuracy	61.5%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.55:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	81.9%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.55:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 18
Accuracy	86.2%
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 19
Accuracy	35.8%
Cost	13120

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

Alternative 20
Accuracy	34.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023140 
(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))))