Average Error: 19.8 → 0.7
Time: 4.0s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\frac{\sqrt{x} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)\right)}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\frac{\sqrt{x} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)\right)}{1}}
double code(double x) {
	return ((double) (((double) (1.0 / ((double) sqrt(x)))) - ((double) (1.0 / ((double) sqrt(((double) (x + 1.0))))))));
}
↓
double code(double x) {
	return ((double) (1.0 / ((double) (((double) (((double) sqrt(x)) * ((double) (((double) (((double) sqrt(((double) (x + 1.0)))) * ((double) sqrt(x)))) + ((double) (x + 1.0)))))) / 1.0))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.5

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

  7. Simplified19.2

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

  8. Taylor expanded around 0 0.4

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

  10. Applied associate-/l*0.8

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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