Main:z from

Average Error: 5.2 → 0.1

Time: 16.4s

Precision: binary64

Cost: 53184

Math

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

↓

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

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
 (+
  (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))
  (+
   (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
    (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

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) {
	return (1.0 / (sqrt(1.0 + t) + sqrt(t))) + ((1.0 / (sqrt(1.0 + z) + sqrt(z))) + ((1.0 / (sqrt(1.0 + y) + sqrt(y))) + (1.0 / (sqrt(1.0 + x) + sqrt(x)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.2
Target	1.5
Herbie	0.1

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

Alternatives

Alternative 1
Error	1.5
Cost	53056

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

Alternative 2
Error	2.8
Cost	52928

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

Alternative 3
Error	2.8
Cost	52928

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

Alternative 4
Error	2.9
Cost	53442

\[\begin{array}{l} \mathbf{if}\;x \leq 3.129973877463878 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \mathbf{elif}\;x \leq 4.962894724549568 \cdot 10^{+38}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \end{array}\]

Alternative 5
Error	3.3
Cost	66177

\[\begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.9999999999999993:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 6
Error	4.5
Cost	66049

\[\begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.9999999999999997:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 7
Error	15.2
Cost	41863

\[\begin{array}{l} \mathbf{if}\;z \leq 1.749209710265025 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.4186793747947015 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \mathbf{elif}\;z \leq 2.6390803519930545 \cdot 10^{+114}:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 3.0689642412072 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \mathbf{elif}\;z \leq 1.040237809890818 \cdot 10^{+264}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.1242072471687015 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \mathbf{elif}\;z \leq 1.2451061023175456 \cdot 10^{+300}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array}\]

Alternative 8
Error	15.1
Cost	40514

\[\begin{array}{l} \mathbf{if}\;t \leq 4.261511489370577 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;t \leq 6.427187126760599 \cdot 10^{+236}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 9
Error	15.3
Cost	40386

\[\begin{array}{l} \mathbf{if}\;t \leq 4.261511489370577 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;t \leq 6.984812305218358 \cdot 10^{+238}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 10
Error	15.5
Cost	40386

\[\begin{array}{l} \mathbf{if}\;t \leq 4.261511489370577 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;t \leq 6.286786470220514 \cdot 10^{+236}:\\ \;\;\;\;\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 11
Error	15.7
Cost	40386

\[\begin{array}{l} \mathbf{if}\;t \leq 4.261511489370577 \cdot 10^{-23}:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;t \leq 1.512039287113479 \cdot 10^{+244}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array}\]

Alternative 12
Error	16.1
Cost	41863

\[\begin{array}{l} \mathbf{if}\;z \leq 1.749209710265025 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 8.941941906127086 \cdot 10^{+59}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 3.884666491496153 \cdot 10^{+112}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.8496303324481346 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \mathbf{elif}\;z \leq 8.093901973367772 \cdot 10^{+263}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 8.878949147017514 \cdot 10^{+284}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.0629028857001353 \cdot 10^{+300}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array}\]

Alternative 13
Error	16.1
Cost	41863

\[\begin{array}{l} \mathbf{if}\;z \leq 1.749209710265025 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.3203384027605776 \cdot 10^{+60}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.0453470670416284 \cdot 10^{+111}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.922741635367823 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.8204182951657183 \cdot 10^{+264}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.3948658972908794 \cdot 10^{+286}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 9.034750711599012 \cdot 10^{+299}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array}\]

Alternative 14
Error	16.0
Cost	40397

\[\begin{array}{l} \mathbf{if}\;z \leq 1.749209710265025 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;z \leq 5.210283662573563 \cdot 10^{+59} \lor \neg \left(z \leq 4.215983847752865 \cdot 10^{+111}\right) \land z \leq 3.55821439403702 \cdot 10^{+218}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \end{array}\]

Alternative 15
Error	16.8
Cost	40265

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1531057024495583 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1484045432768985 \cdot 10^{+175} \lor \neg \left(x \leq 7.940079885939886 \cdot 10^{+225}\right):\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 16
Error	20.1
Cost	39937

\[\begin{array}{l} \mathbf{if}\;x \leq 8.073342772398961:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 17
Error	42.0
Cost	64

\[1\]

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. Using strategy rm

3. Applied flip--_binary64_14716 5.1

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

4. Simplified 4.0

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

6. Applied flip--_binary64_14716 3.9

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

7. Simplified 2.8

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

8. Simplified 2.8

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

10. Applied flip--_binary64_14716 2.7

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

11. Simplified 1.5

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

13. Applied flip--_binary64_14716 1.4

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

14. Simplified 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2021044 
(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))))