Main:z from

Average Error: 5.2 → 1.3

Time: 24.1s

Precision: binary64

Cost: 66112

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{y} + \sqrt{1 + y}\\ \frac{1 + t_1 \cdot \frac{1}{\sqrt{x} + \sqrt{1 + x}}}{t_1} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \end{array} \]

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 y) (sqrt (+ 1.0 y)))))
   (+
    (/ (+ 1.0 (* t_1 (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))))) t_1)
    (+ (- (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(y) + sqrt((1.0 + y));
	return ((1.0 + (t_1 * (1.0 / (sqrt(x) + sqrt((1.0 + x)))))) / t_1) + ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t)));
}

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
    t_1 = sqrt(y) + sqrt((1.0d0 + y))
    code = ((1.0d0 + (t_1 * (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))))) / t_1) + ((sqrt((1.0d0 + z)) - sqrt(z)) + (sqrt((1.0d0 + t)) - sqrt(t)))
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(y) + Math.sqrt((1.0 + y));
	return ((1.0 + (t_1 * (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))))) / t_1) + ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
}

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(y) + math.sqrt((1.0 + y))
	return ((1.0 + (t_1 * (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))))) / t_1) + ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (math.sqrt((1.0 + t)) - math.sqrt(t)))

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(y) + sqrt(Float64(1.0 + y)))
	return Float64(Float64(Float64(1.0 + Float64(t_1 * Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))) / t_1) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))))
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 = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt((1.0 + y));
	tmp = ((1.0 + (t_1 * (1.0 / (sqrt(x) + sqrt((1.0 + x)))))) / t_1) + ((sqrt((1.0 + z)) - sqrt(z)) + (sqrt((1.0 + t)) - sqrt(t)));
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[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + N[(t$95$1 * N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.2
Target	0.4
Herbie	1.3

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

Derivation

1. Initial program 5.2

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

2. Simplified 5.2

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

   [Start] 5.2	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 5.2	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
   associate-+l- [=>] 5.2	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate--r- [=>] 5.9	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [<=] 5.9	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate-+l- [=>] 5.2	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 5.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [=>] 5.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [=>] 5.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [<=] 5.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 5.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

3. Applied egg-rr 5.2

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

4. Applied egg-rr 1.3

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

5. Applied egg-rr 1.3

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

6. Simplified 1.3

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

   [Start] 1.3	\[ \frac{\left(\left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right) - \left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-commutative [=>] 1.3	\[ \frac{\color{blue}{\frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)} \cdot \left(\left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right) - \left(\sqrt{1 + y} + \sqrt{y}\right)\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   sub-neg [=>] 1.3	\[ \frac{\frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)} \cdot \color{blue}{\left(\left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right) + \left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right)\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   distribute-rgt-in [=>] 1.3	\[ \frac{\color{blue}{\left(-\left(\sqrt{1 + x} + \sqrt{x}\right)\right) \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)} + \left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   rgt-mult-inverse [=>] 1.3	\[ \frac{\color{blue}{1} + \left(-\left(\sqrt{1 + y} + \sqrt{y}\right)\right) \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 1.3	\[ \frac{1 + \left(-\color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}\right) \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   distribute-neg-in [=>] 1.3	\[ \frac{1 + \color{blue}{\left(\left(-\sqrt{y}\right) + \left(-\sqrt{1 + y}\right)\right)} \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   sub-neg [<=] 1.3	\[ \frac{1 + \color{blue}{\left(\left(-\sqrt{y}\right) - \sqrt{1 + y}\right)} \cdot \frac{1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   neg-mul-1 [=>] 1.3	\[ \frac{1 + \left(\left(-\sqrt{y}\right) - \sqrt{1 + y}\right) \cdot \frac{1}{\color{blue}{-1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-/r* [=>] 1.3	\[ \frac{1 + \left(\left(-\sqrt{y}\right) - \sqrt{1 + y}\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 1.3	\[ \frac{1 + \left(\left(-\sqrt{y}\right) - \sqrt{1 + y}\right) \cdot \frac{\color{blue}{-1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 1.3	\[ \frac{1 + \left(\left(-\sqrt{y}\right) - \sqrt{1 + y}\right) \cdot \frac{-1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

7. Final simplification 1.3

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

Alternatives

Alternative 1
Error	2.6
Cost	52800

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

Alternative 2
Error	4.4
Cost	40008

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(t_2 - \sqrt{z}\right) + t_3\right) + \left(1 + \frac{1}{\sqrt{y} + t_1}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;t_4 + \left(\left(t_1 - \sqrt{y}\right) + \left(t_3 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_3 + \frac{1}{t_2 + \sqrt{z}}\right) + \left(\left(1 + t_4\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 3
Error	4.4
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;x \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right) + \left(1 + \frac{1}{\sqrt{y} + t_2}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_2 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	4.5
Cost	39872

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

Alternative 5
Error	5.9
Cost	39752

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;2 + \left(t_2 + \left(t_3 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(t_1 + \left(\left(t_2 - \sqrt{y}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{y}\right) + \left(\left(t_3 - \sqrt{t}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 6
Error	5.4
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right) + \left(\left(1 + t_2\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_2 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	6.2
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 17000000:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 - \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	6.0
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;2 + \left(t_2 + \left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(t_1 + \left(\left(t_2 - \sqrt{y}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\sqrt{y} + t_1}\\ \end{array} \]

Alternative 9
Error	9.0
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(t_1 + \left(\left(\sqrt{1 + z} - \sqrt{y}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\sqrt{y} + t_1}\\ \end{array} \]

Alternative 10
Error	11.5
Cost	13508

\[\begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ \mathbf{if}\;z \leq 0.55:\\ \;\;\;\;t_1 + 2\\ \mathbf{else}:\\ \;\;\;\;1 + t_1\\ \end{array} \]

Alternative 11
Error	25.3
Cost	13380

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

Alternative 12
Error	22.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	21.8
Cost	13376

\[1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}} \]

Alternative 14
Error	41.3
Cost	13248

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

Alternative 15
Error	41.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023034 
(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))))