Average Error: 19.8 → 0.8
Time: 5.3s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\frac{1 + x}{1} \cdot \left(\frac{x}{1} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\frac{1 + x}{1} \cdot \left(\frac{x}{1} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)\right)}
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 / ((double) ((((double) (1.0 + x)) / 1.0) * ((double) ((x / 1.0) * ((double) ((1.0 / ((double) sqrt(x))) + (1.0 / ((double) sqrt(((double) (1.0 + x))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.8

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

  3. Applied flip--19.8

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

  4. Simplified19.9

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

  5. Simplified19.9

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

  7. Applied frac-sub19.3

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

  8. Simplified19.3

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

  10. Applied div-inv19.3

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

  11. Applied associate-/l*19.3

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

  12. Simplified19.2

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

  13. Taylor expanded around 0 0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2020196 
(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)))))