Average Error: 19.1 → 0.4
Time: 2.9s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x + \sqrt{x} \cdot \sqrt{1 + x}}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x + \sqrt{x} \cdot \sqrt{1 + x}}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/
  (/ 1.0 (+ x (* (sqrt x) (sqrt (+ 1.0 x)))))
  (* (sqrt (sqrt (+ 1.0 x))) (sqrt (sqrt (+ 1.0 x))))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt(x + 1.0));
}
↓
double code(double x) {
	return (1.0 / (x + (sqrt(x) * sqrt(1.0 + x)))) / (sqrt(sqrt(1.0 + x)) * sqrt(sqrt(1.0 + x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.1
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.1

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

  3. Applied frac-sub_binary6419.1

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

  4. Simplified19.1

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

  5. Simplified19.1

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

  7. Applied flip--_binary6418.9

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

  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 associate-/r*_binary640.4

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

  11. Simplified0.3

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

  13. Applied add-sqr-sqrt_binary640.3

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

  14. Applied sqrt-prod_binary640.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020219 
(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)))))