Average Error: 20.1 → 0.5
Time: 6.2s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[1 \cdot \left(\frac{\frac{1}{x}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}} \cdot \frac{1}{1 + x}\right)\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
1 \cdot \left(\frac{\frac{1}{x}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}} \cdot \frac{1}{1 + x}\right)
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) (1.0 / x)) / ((double) (((double) (1.0 / ((double) sqrt(((double) (1.0 + x)))))) + ((double) (1.0 / ((double) sqrt(x)))))))) * ((double) (1.0 / ((double) (1.0 + x))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied flip--20.1

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

  4. Simplified20.2

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

  5. Simplified20.2

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

  7. Applied frac-sub19.5

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

  8. Applied associate-/l/19.5

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

  9. Simplified19.5

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

  10. Taylor expanded around 0 0.8

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

  12. Applied associate-*l/0.8

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

  13. Applied associate-*r/0.8

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

  14. Applied associate-/r/0.8

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020184 
(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)))))