Average Error: 19.8 → 5.5
Time: 10.7s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}
double code(double x) {
	return ((double) ((1.0 / ((double) sqrt(x))) - (1.0 / ((double) sqrt(((double) (x + 1.0)))))));
}
↓
double code(double x) {
	return ((((double) (1.0 * 1.0)) / ((double) (x * ((double) (1.0 + x))))) / ((double) (((double) pow(x, -0.5)) + (1.0 / ((double) sqrt(((double) (1.0 + x))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie5.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Simplified19.8

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

  5. Simplified19.8

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

  7. Applied div-inv19.8

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

  8. Applied div-inv19.8

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

  9. Applied distribute-lft-out19.8

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

  10. Applied associate-/r*19.8

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

  11. Simplified19.8

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

  13. Applied pow1/219.8

    \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\frac{1}{\color{blue}{{x}^{0.5}}} + \frac{1}{\sqrt{1 + x}}}\]

  14. Applied pow-flip19.8

    \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{{x}^{\left(-0.5\right)}} + \frac{1}{\sqrt{1 + x}}}\]

  15. Simplified19.8

    \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{\color{blue}{-0.5}} + \frac{1}{\sqrt{1 + x}}}\]
  16. Using strategy rm

  17. Applied frac-sub19.2

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\]

  18. Simplified5.5

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot 1 + 0}}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\]

  19. Final simplification5.5

    \[\leadsto \frac{\frac{1 \cdot 1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\]

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)))))