Main:z from

Average Error: 5.6 → 1.0

Time: 25.6s

Precision: binary64

Cost: 66116

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.001:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]

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 (sqrt (+ x 1.0))))
   (if (<= (- t_1 (sqrt x)) 0.001)
     (/ 1.0 (+ t_1 (sqrt x)))
     (+
      (- (+ t_1 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))) (sqrt x))
      (+
       (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
       (- (sqrt (+ 1.0 t)) (sqrt t)))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double tmp;
	if ((t_1 - sqrt(x)) <= 0.001) {
		tmp = 1.0 / (t_1 + sqrt(x));
	} else {
		tmp = ((t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x)) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - 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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if ((t_1 - sqrt(x)) <= 0.001d0) then
        tmp = 1.0d0 / (t_1 + sqrt(x))
    else
        tmp = ((t_1 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) - sqrt(x)) + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.001) {
		tmp = 1.0 / (t_1 + Math.sqrt(x));
	} else {
		tmp = ((t_1 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) - Math.sqrt(x)) + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.001:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	else:
		tmp = ((t_1 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) - math.sqrt(x)) + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t)))
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.001)
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) - sqrt(x)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.001)
		tmp = 1.0 / (t_1 + sqrt(x));
	else
		tmp = ((t_1 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) - sqrt(x)) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	5.6
Target	0.4
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1e-3

   1. Initial program 57.9

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

   2. Simplified 60.9

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

      [Start] 57.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 57.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 57.9	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 57.9	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 57.9	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 57.9	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 57.9	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 57.9	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in z around inf 59.3

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

   4. Simplified 57.9

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

      [Start] 59.3	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} + \sqrt{t}\right) - \sqrt{1 + t}\right)\right) \]
      associate--l+ [=>] 57.9	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{t} - \sqrt{1 + t}\right)\right)}\right) \]
      +-commutative [=>] 57.9	\[ \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{t} - \sqrt{\color{blue}{t + 1}}\right)\right)\right) \]

   5. Taylor expanded in y around inf 59.3

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

   6. Taylor expanded in t around inf 57.9

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   7. Applied egg-rr 9.2

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

   8. Simplified 9.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
      Proof

      [Start] 9.2	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [=>] 9.2	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      +-inverses [=>] 9.2	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      metadata-eval [=>] 9.2	\[ \color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      *-lft-identity [=>] 9.2	\[ \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

   if 1e-3 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 2.1

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

   2. Simplified 2.1

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

      [Start] 2.1	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.1	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 2.1	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 2.8	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 2.8	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 2.1	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 1.3

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

   4. Simplified 1.3

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

      [Start] 1.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 1.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 1.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 1.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 1.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 0.8

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

   6. Taylor expanded in y around 0 0.4

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

   7. Applied egg-rr 0.4

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.001:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	66244

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

Alternative 2
Error	1.0
Cost	66116

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.001:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + t}\\ \mathbf{if}\;y \leq 6 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(t_2 - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(1 + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+33}:\\ \;\;\;\;t_1 - \left(\left(\sqrt{x} + \left(\sqrt{t} - t_2\right)\right) + \frac{-1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 34500000000000:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 5
Error	1.9
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 34500000000000:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 6
Error	5.0
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + \frac{1}{t_1 + \sqrt{t}}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\left(\sqrt{x} + \left(\sqrt{t} - t_1\right)\right) + \frac{-1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \]

Alternative 7
Error	2.7
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 34500000000000:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 8
Error	5.8
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + \frac{1}{t_1 + \sqrt{t}}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 9
Error	7.0
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{1 + z} + \left(\left(t_1 + \left(1 - \sqrt{y}\right)\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{y} + \left(\sqrt{x} - t_1\right)\right)\\ \end{array} \]

Alternative 10
Error	6.5
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{y} + \left(\sqrt{x} - \sqrt{1 + y}\right)\right)\\ \end{array} \]

Alternative 11
Error	9.1
Cost	26436

\[\begin{array}{l} \mathbf{if}\;z \leq 0.9:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \left(\sqrt{y} + \left(\sqrt{x} - \sqrt{1 + y}\right)\right)\\ \end{array} \]

Alternative 12
Error	24.8
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;y \leq 2:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	9.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 0.34:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14
Error	23.0
Cost	13248

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

Alternative 15
Error	41.3
Cost	13120

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

Alternative 16
Error	41.7
Cost	6848

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

Alternative 17
Error	42.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023067 
(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))))