Average Error: 19.5 → 0.3
Time: 5.4s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{1 + \left(x + \sqrt{x} \cdot \sqrt{1 + x}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{1 + \left(x + \sqrt{x} \cdot \sqrt{1 + x}\right)}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (* (/ 1.0 (sqrt x)) (/ 1.0 (+ 1.0 (+ x (* (sqrt 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 / sqrt(x)) * (1.0 / (1.0 + (x + (sqrt(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

Original19.5
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied frac-sub_binary6419.5

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

  4. Simplified19.5

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

  5. Simplified19.5

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

  7. Applied flip--_binary6419.4

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

  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

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

  13. Applied *-un-lft-identity_binary640.3

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

  14. Applied sqrt-prod_binary640.3

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

  15. Applied add-sqr-sqrt_binary640.4

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

  16. Applied distribute-rgt-out_binary640.4

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

  17. Applied add-cube-cbrt_binary640.4

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}{\sqrt{1} \cdot \sqrt{1 + x}}\]

  18. Applied times-frac_binary640.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{1} \cdot \sqrt{1 + x}}\]

  19. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x}}}{\sqrt{1}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{1 + x}}}\]

  20. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{x}}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{1 + x}}\]

  21. Simplified0.3

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

  22. Final simplification0.3

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

Reproduce

herbie shell --seed 2021174 
(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)))))