Main:z from

Average Accuracy: 91.7% → 99.6%

Time: 29.1s

Precision: binary64

Cost: 53060

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;y \leq 100:\\ \;\;\;\;\left(t_3 + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\frac{1}{t_2 + \sqrt{z}} + \frac{1}{t_4 + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{x} + t_3}\right) + \left(\left(t_4 - \sqrt{t}\right) + \left(t_2 - \sqrt{z}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= y 100.0)
     (+
      (+ t_3 (- (- t_1 (sqrt y)) (sqrt x)))
      (+ (/ 1.0 (+ t_2 (sqrt z))) (/ 1.0 (+ t_4 (sqrt t)))))
     (+
      (+ (/ 1.0 (+ t_1 (sqrt y))) (/ 1.0 (+ (sqrt x) t_3)))
      (+ (- t_4 (sqrt t)) (- t_2 (sqrt z)))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if (y <= 100.0) {
		tmp = (t_3 + ((t_1 - sqrt(y)) - sqrt(x))) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (t_4 + sqrt(t))));
	} else {
		tmp = ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt(x) + t_3))) + ((t_4 - sqrt(t)) + (t_2 - sqrt(z)));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + x))
    t_4 = sqrt((1.0d0 + t))
    if (y <= 100.0d0) then
        tmp = (t_3 + ((t_1 - sqrt(y)) - sqrt(x))) + ((1.0d0 / (t_2 + sqrt(z))) + (1.0d0 / (t_4 + sqrt(t))))
    else
        tmp = ((1.0d0 / (t_1 + sqrt(y))) + (1.0d0 / (sqrt(x) + t_3))) + ((t_4 - sqrt(t)) + (t_2 - sqrt(z)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if (y <= 100.0) {
		tmp = (t_3 + ((t_1 - Math.sqrt(y)) - Math.sqrt(x))) + ((1.0 / (t_2 + Math.sqrt(z))) + (1.0 / (t_4 + Math.sqrt(t))));
	} else {
		tmp = ((1.0 / (t_1 + Math.sqrt(y))) + (1.0 / (Math.sqrt(x) + t_3))) + ((t_4 - Math.sqrt(t)) + (t_2 - Math.sqrt(z)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + x))
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if y <= 100.0:
		tmp = (t_3 + ((t_1 - math.sqrt(y)) - math.sqrt(x))) + ((1.0 / (t_2 + math.sqrt(z))) + (1.0 / (t_4 + math.sqrt(t))))
	else:
		tmp = ((1.0 / (t_1 + math.sqrt(y))) + (1.0 / (math.sqrt(x) + t_3))) + ((t_4 - math.sqrt(t)) + (t_2 - math.sqrt(z)))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

↓

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (y <= 100.0)
		tmp = Float64(Float64(t_3 + Float64(Float64(t_1 - sqrt(y)) - sqrt(x))) + Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(1.0 / Float64(t_4 + sqrt(t)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(1.0 / Float64(sqrt(x) + t_3))) + Float64(Float64(t_4 - sqrt(t)) + Float64(t_2 - sqrt(z))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + x));
	t_4 = sqrt((1.0 + t));
	tmp = 0.0;
	if (y <= 100.0)
		tmp = (t_3 + ((t_1 - sqrt(y)) - sqrt(x))) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (t_4 + sqrt(t))));
	else
		tmp = ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt(x) + t_3))) + ((t_4 - sqrt(t)) + (t_2 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 100.0], N[(N[(t$95$3 + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	91.7%
Target	99.4%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if y < 100

   1. Initial program 97.4%

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

   2. Simplified 97.4%

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

      [Start] 97.4	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 97.4	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 97.4	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 97.4	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 97.4	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 97.4	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 99.2%

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

      [Start] 97.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 97.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 75.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 98.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   4. Simplified 99.2%

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

      [Start] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 99.9%

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

      [Start] 99.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      div-inv [=>] 99.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
      add-sqr-sqrt [<=] 58.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      add-sqr-sqrt [<=] 99.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      associate--l+ [=>] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 + \left(t - t\right)\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

   6. Simplified 99.9%

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

      [Start] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\left(t - t\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 99.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   if 100 < y

   1. Initial program 78.8%

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

   2. Simplified 78.8%

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

      [Start] 78.8	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 78.8	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 78.8	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 75.0	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 75.0	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 78.8	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 82.4%

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

      [Start] 78.8	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 79.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 79.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 49.0	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 80.6	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\left(1 + y\right) - \color{blue}{y}\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(1 + \left(y - y\right)\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   4. Simplified 82.4%

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

      [Start] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 83.4%

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

      [Start] 82.4	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 83.4	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 83.4	\[ \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Applied egg-rr 84.6%

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

      [Start] 83.4	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      flip-- [=>] 83.9	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      div-inv [=>] 83.9	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 83.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      add-sqr-sqrt [<=] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate--l+ [=>] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Simplified 99.1%

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

      [Start] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-commutative [=>] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \left(x + \left(1 - x\right)\right)}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*l/ [=>] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1 \cdot \left(x + \left(1 - x\right)\right)}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 84.6	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 99.1	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 99.1	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 99.1	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 99.1	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 100:\\ \;\;\;\;\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	53056

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

Alternative 2
Accuracy	98.3%
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{1}{t_3 + \sqrt{z}} + t_1\right) + \left(1 + \frac{1}{t_2 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(t_3 - \sqrt{z}\right)\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(t_2 - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	98.0%
Cost	52928

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

Alternative 4
Accuracy	94.2%
Cost	40136

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := t_1 - \sqrt{t}\\ t_3 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ t_4 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \mathbf{if}\;x \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;\left(t_4 + t_2\right) + \left(1 + t_3\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(t_3 + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(t_4 + \frac{1}{t_1 + \sqrt{t}}\right)\\ \end{array} \]

Alternative 5
Accuracy	94.2%
Cost	40008

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ t_3 := \frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\\ \mathbf{if}\;x \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t_3 + \left(1 + t_2\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(t_2 + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + t_3\\ \end{array} \]

Alternative 6
Accuracy	93.0%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;\left(1 + t_2\right) - \left(\left(\sqrt{t} - t_1\right) + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(t_2 + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	93.6%
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\right) + \left(\left(1 + t_2\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{t_2 + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	91.3%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;t \leq 24000000000000:\\ \;\;\;\;1 + \left(\left(t_2 - \sqrt{z}\right) + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(1 + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	92.3%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	92.8%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 6.1 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(t_2 - \sqrt{y}\right)\right) - \left(\left(\sqrt{t} - t_1\right) + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{t_2 + \sqrt{y}} + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	92.8%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\left(t_2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(1 + t_1\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{t_1 + \sqrt{y}} + \left(t_2 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	91.3%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;t \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + t_1\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(1 + t_1\right)\\ \end{array} \]

Alternative 13
Accuracy	86.3%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(t_1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{y}\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{y}}\\ \end{array} \]

Alternative 14
Accuracy	91.1%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 15
Accuracy	86.0%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \left(1 + y \cdot 0.5\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 16
Accuracy	84.7%
Cost	20292

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \left(1 + y \cdot 0.5\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 17
Accuracy	84.6%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18
Accuracy	84.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19
Accuracy	64.5%
Cost	13248

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

Alternative 20
Accuracy	34.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023139 
(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))))