Average Error: 29.6 → 0.2

Time: 13.0s

Precision: 64

Internal Precision: 1344

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

↓

\[\frac{1}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	29.6
Target	0.2
Herbie	0.2

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

Derivation

Initial program 29.6
\[\sqrt{x + 1} - \sqrt{x}\]
Using strategy rm
Applied flip--29.4
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]
Taylor expanded around 0 0.2
\[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
Using strategy rm
Applied add-sqr-sqrt0.3
\[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}} + \sqrt{x}}\]
Applied fma-def0.2
\[\leadsto \frac{1}{\color{blue}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}}\]
Final simplification0.2
\[\leadsto \frac{1}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.2	0.2	0.0	0.2	0%

herbie shell --seed 2018295 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))