Average Error: 20.0 → 0.3
Time: 7.0s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{\frac{1}{1 + x}}{\sqrt{x} + \frac{x}{\sqrt{1 + x}}} \]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{1 + x}}{\sqrt{x} + \frac{x}{\sqrt{1 + x}}}

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ 1.0 x)) (+ (sqrt x) (/ x (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 / (1.0 + x)) / (sqrt(x) + (x / sqrt(1.0 + x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 20.0

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

  3. Applied flip--_binary6420.0

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

  4. Simplified20.0

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

  5. Simplified20.0

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

  7. Applied frac-sub_binary6419.4

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

  8. Simplified6.0

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

  9. Simplified6.0

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

  11. Applied distribute-rgt1-in_binary646.0

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

  12. Applied *-un-lft-identity_binary646.0

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

  13. Applied times-frac_binary645.5

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

  14. Applied associate-/l*_binary640.4

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

  15. Simplified0.4

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

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

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

  18. Applied associate-*l*_binary640.4

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2021206 
(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)))))