Average Error: 19.8 → 0.5
Time: 10.9s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\frac{x}{1}}}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{\frac{x + 1}{1}}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\frac{x}{1}}}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{\frac{x + 1}{1}}}}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/
  (/ 1.0 (/ x 1.0))
  (/ (+ (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))) (/ 1.0 (/ (+ x 1.0) 1.0)))))
double code(double x) {
	return ((double) ((1.0 / ((double) sqrt(x))) - (1.0 / ((double) sqrt(((double) (x + 1.0)))))));
}
↓
double code(double x) {
	return ((1.0 / (x / 1.0)) / (((double) ((1.0 / ((double) sqrt(x))) + (1.0 / ((double) sqrt(((double) (x + 1.0))))))) / (1.0 / (((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
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program Error: 19.8 bits

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

  3. Applied flip--Error: 19.8 bits

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

  4. SimplifiedError: 19.8 bits

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

  5. SimplifiedError: 19.8 bits

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

  7. Applied frac-subError: 19.2 bits

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

  8. SimplifiedError: 19.2 bits

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

  9. Taylor expanded around 0 Error: 5.6 bits

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

  11. Applied *-un-lft-identityError: 5.6 bits

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

  12. Applied times-fracError: 5.1 bits

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

  13. Applied associate-/l*Error: 0.5 bits

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

  14. Final simplificationError: 0.5 bits

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

Reproduce

herbie shell --seed 2020205 
(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)))))