Average Error: 19.6 → 0.2
Time: 8.7s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[{x}^{-0.5} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{x}^{-0.5} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (* (pow x -0.5) (/ 1.0 (+ (+ x 1.0) (* (sqrt x) (sqrt (+ x 1.0)))))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt(x + 1.0));
}
↓
double code(double x) {
	return pow(x, -0.5) * (1.0 / ((x + 1.0) + (sqrt(x) * sqrt(x + 1.0))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.6
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub_binary6419.6

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

  4. Simplified19.6

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

  5. Simplified19.6

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

  7. Applied flip--_binary6419.3

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity_binary640.4

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

  11. Applied add-sqr-sqrt_binary640.4

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

  12. Applied times-frac_binary640.4

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

  13. Applied times-frac_binary640.4

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

  14. Simplified0.4

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

  15. Simplified0.3

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

  17. Applied pow1/2_binary640.3

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

  18. Applied pow-flip_binary640.2

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

  19. Simplified0.2

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

  20. Final simplification0.2

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

Reproduce

herbie shell --seed 2020268 
(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)))))