Main:z from

Average Error: 5.5 → 1.6

Time: 27.8s

Precision: binary64

Cost: 53060

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + t}\\ \mathbf{if}\;x \leq 10^{-14}:\\ \;\;\;\;\left(t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(t_1 + \frac{1}{t_3 + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}} + \left(t_1 + \left(t_3 - \sqrt{t}\right)\right)\\ \end{array} \]

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 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 t))))
   (if (<= x 1e-14)
     (+
      (+ t_2 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (sqrt x)))
      (+ t_1 (/ 1.0 (+ t_3 (sqrt t)))))
     (+ (/ 1.0 (+ t_2 (sqrt x))) (+ t_1 (- t_3 (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 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + t));
	double tmp;
	if (x <= 1e-14) {
		tmp = (t_2 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x))) + (t_1 + (1.0 / (t_3 + sqrt(t))));
	} else {
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_1 + (t_3 - 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) :: tmp
    t_1 = 1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + t))
    if (x <= 1d-14) then
        tmp = (t_2 + ((sqrt((1.0d0 + y)) - sqrt(y)) - sqrt(x))) + (t_1 + (1.0d0 / (t_3 + sqrt(t))))
    else
        tmp = (1.0d0 / (t_2 + sqrt(x))) + (t_1 + (t_3 - 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 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + t));
	double tmp;
	if (x <= 1e-14) {
		tmp = (t_2 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - Math.sqrt(x))) + (t_1 + (1.0 / (t_3 + Math.sqrt(t))));
	} else {
		tmp = (1.0 / (t_2 + Math.sqrt(x))) + (t_1 + (t_3 - 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 = 1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + t))
	tmp = 0
	if x <= 1e-14:
		tmp = (t_2 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - math.sqrt(x))) + (t_1 + (1.0 / (t_3 + math.sqrt(t))))
	else:
		tmp = (1.0 / (t_2 + math.sqrt(x))) + (t_1 + (t_3 - 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(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (x <= 1e-14)
		tmp = Float64(Float64(t_2 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - sqrt(x))) + Float64(t_1 + Float64(1.0 / Float64(t_3 + sqrt(t)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + Float64(t_1 + Float64(t_3 - 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 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + t));
	tmp = 0.0;
	if (x <= 1e-14)
		tmp = (t_2 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x))) + (t_1 + (1.0 / (t_3 + sqrt(t))));
	else
		tmp = (1.0 / (t_2 + sqrt(x))) + (t_1 + (t_3 - 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[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1e-14], N[(N[(t$95$2 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	5.5
Target	0.4
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999999e-15

   1. Initial program 2.0

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

   2. Simplified 2.0

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

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

   3. Applied egg-rr 1.2

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

   4. Simplified 1.2

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

      [Start] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 1.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 1.1

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

   6. Simplified 0.9

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

      [Start] 1.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-commutative [=>] 1.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\left(1 + t\right) - t\right)}\right) \]
      associate--l+ [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}\right) \]
      distribute-rgt-in [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)}\right) \]
      +-inverses [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{0} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [<=] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{\left(z - z\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      distribute-rgt-out [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(1 + \left(z - z\right)\right)}\right) \]
      +-commutative [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(\left(z - z\right) + 1\right)}\right) \]
      +-inverses [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\color{blue}{0} + 1\right)\right) \]
      metadata-eval [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   if 9.99999999999999999e-15 < x

   1. Initial program 47.1

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

   2. Simplified 47.1

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

      [Start] 47.1	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 47.1	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 47.1	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 54.5	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 54.5	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 47.1	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 47.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 47.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 47.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 47.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 47.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Taylor expanded in y around inf 48.7

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

   4. Applied egg-rr 9.6

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

   5. Simplified 9.6

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

      [Start] 9.6	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 9.6	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 9.6	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 9.6	\[ \color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 9.6	\[ \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Applied egg-rr 9.6

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

   7. Simplified 9.6

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

      [Start] 9.6	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 9.6	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 9.6	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 9.6	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 9.6	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.6

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

Alternatives

Alternative 1
Error	1.6
Cost	66244

\[\begin{array}{l} t_1 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;\left(t_2 - \sqrt{y}\right) + \left(t_3 - \sqrt{x}\right) \leq 1.000000015:\\ \;\;\;\;\frac{1}{t_3 + \sqrt{x}} + \left(t_1 + \left(t_4 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \frac{1}{t_4 + \sqrt{t}}\right) + \left(\left(1 + t_2\right) - \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 4
Error	1.9
Cost	39876

\[\begin{array}{l} t_1 := \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}} + t_1\\ \end{array} \]

Alternative 5
Error	2.7
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 6
Error	2.2
Cost	27208

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.02 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+30}:\\ \;\;\;\;t_1 + \frac{1}{\frac{1}{\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 7
Error	3.1
Cost	26824

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{z}} + \left(1 + \sqrt{1 + y}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 8
Error	3.1
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 9
Error	6.6
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-27}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 10
Error	22.0
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-21} \lor \neg \left(x \leq 1.3 \cdot 10^{+14}\right):\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 11
Error	9.8
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+36}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 12
Error	6.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 13
Error	40.9
Cost	13120

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

Alternative 14
Error	41.3
Cost	6848

\[\left(1 + x \cdot 0.5\right) - \sqrt{x} \]

Alternative 15
Error	41.7
Cost	6592

\[1 - \sqrt{x} \]

Alternative 16
Error	41.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))