Average Error: 29.4 → 0.3
Time: 4.1s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \sqrt{x}}
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (sqrt (sqrt (+ 1.0 x))) (sqrt (sqrt (+ 1.0 x)))) (sqrt x))))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	return 1.0 / ((sqrt(sqrt(1.0 + x)) * sqrt(sqrt(1.0 + x))) + sqrt(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--_binary64_140029.2

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt_binary64_14460.3

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

  7. Simplified0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020299 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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