Average Error: 20.2 → 0.6
Time: 9.1s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x} \cdot \left(1 + x\right) + x \cdot \sqrt{1 + x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x} \cdot \left(1 + x\right) + x \cdot \sqrt{1 + x}}
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (sqrt x) (+ 1.0 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 / ((sqrt(x) * (1.0 + 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

Original20.2
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied frac-sub_binary6420.2

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

  4. Simplified20.2

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

  5. Simplified20.2

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

  7. Applied flip--_binary6420.0

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

  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-/l/_binary640.8

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

  11. Simplified0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2020277 
(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)))))