Average Error: 20.2 → 0.3
Time: 5.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(-1\right) \cdot \frac{1}{\sqrt{x + 1}}}{-\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(-1\right) \cdot \frac{1}{\sqrt{x + 1}}}{-\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)}
double code(double x) {
	return ((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0))));
}
↓
double code(double x) {
	return ((-1.0 * (1.0 / sqrt((x + 1.0)))) / -fma(sqrt((x + 1.0)), sqrt(x), x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied clear-num20.2

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

  4. Applied frac-sub20.2

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

  5. Simplified20.2

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

  7. Applied flip--19.9

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

  8. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \frac{\sqrt{x + 1}}{1}}\]
  9. Using strategy rm

  10. Applied *-commutative0.4

    \[\leadsto \frac{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\frac{\sqrt{x + 1}}{1} \cdot \sqrt{x}}}\]

  11. Applied div-inv0.4

    \[\leadsto \frac{\color{blue}{\left(1 + 0\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\frac{\sqrt{x + 1}}{1} \cdot \sqrt{x}}\]

  12. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{1 + 0}{\frac{\sqrt{x + 1}}{1}} \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x}}}\]

  13. Simplified0.4

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

  14. Simplified0.3

    \[\leadsto \left(\frac{1}{\sqrt{x + 1}} \cdot 1\right) \cdot \color{blue}{\left(\frac{2}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)} \cdot \frac{1}{2}\right)}\]
  15. Using strategy rm

  16. Applied frac-2neg0.3

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

  17. Applied associate-*l/0.3

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

  18. Applied associate-*r/0.3

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))