Average Error: 29.5 → 0.2
Time: 3.6s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt x) (hypot 1.0 (sqrt x)))))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	return 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 29.5

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

  2. Applied flip--_binary6429.3

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

  3. Simplified0.2

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

  4. Simplified0.2

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

  5. Applied add-sqr-sqrt_binary640.2

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

  6. Applied hypot-1-def_binary640.2

    \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)} + \sqrt{x}} \]

  7. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))