Average Error: 19.3 → 0.2
Time: 10.8s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[{x}^{-0.5} \cdot \frac{1}{1 + \left(x + \sqrt{x + 1} \cdot \sqrt{x}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{x}^{-0.5} \cdot \frac{1}{1 + \left(x + \sqrt{x + 1} \cdot \sqrt{x}\right)}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (* (pow x -0.5) (/ 1.0 (+ 1.0 (+ x (* (sqrt (+ x 1.0)) (sqrt x)))))))
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 / (1.0 + (x + (sqrt(x + 1.0) * sqrt(x)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.6
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied frac-sub_binary6419.3

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

  4. Simplified19.3

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

  5. Simplified19.3

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

  7. Applied flip--_binary6419.2

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

  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 div-inv_binary640.4

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

  11. Applied times-frac_binary640.4

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

  12. Simplified0.3

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

  14. Applied pow1_binary640.3

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

  15. Applied sqrt-pow1_binary640.3

    \[\leadsto \frac{1}{\color{blue}{{x}^{\left(\frac{1}{2}\right)}}} \cdot \frac{1}{1 + \left(x + \sqrt{1 + x} \cdot \sqrt{x}\right)}\]

  16. Applied pow-flip_binary640.2

    \[\leadsto \color{blue}{{x}^{\left(-\frac{1}{2}\right)}} \cdot \frac{1}{1 + \left(x + \sqrt{1 + x} \cdot \sqrt{x}\right)}\]

  17. Simplified0.2

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

  18. Final simplification0.2

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

Reproduce

herbie shell --seed 2020288 
(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)))))