Average Error: 19.5 → 0.4
Time: 58.2s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x}}{\sqrt{1 + x} \cdot \left(1 + \frac{\sqrt{1 + x}}{\sqrt{x}}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x}}{\sqrt{1 + x} \cdot \left(1 + \frac{\sqrt{1 + x}}{\sqrt{x}}\right)}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ (/ 1.0 x) (* (sqrt (+ 1.0 x)) (+ 1.0 (/ (sqrt (+ 1.0 x)) (sqrt 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(1.0 + x) * (1.0 + (sqrt(1.0 + x) / sqrt(x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied flip--_binary6419.5

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

  4. Simplified19.6

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

  5. Simplified19.6

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

  7. Applied frac-sub_binary6419.0

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

  8. Simplified5.6

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

  10. Applied *-un-lft-identity_binary645.6

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

  11. Applied times-frac_binary645.2

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

  12. Applied associate-/l*_binary640.4

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

  13. Simplified0.4

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

  15. Applied add-sqr-sqrt_binary640.4

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

  16. Applied associate-*l*_binary640.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020285 
(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)))))