Average Error: 30.0 → 0.2
Time: 3.2s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\sqrt{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}} \]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}}

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (sqrt (pow (+ (sqrt x) (sqrt (+ x 1.0))) -2.0)))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
↓
double code(double x) {
	return sqrt(pow((sqrt(x) + sqrt((x + 1.0))), -2.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x} \]

  2. Applied egg-rr29.4

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied egg-rr0.3

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{{\left(\sqrt[3]{x + 1}\right)}^{2}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}} \]

  5. Applied egg-rr0.2

    \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}}} \]

  6. Final simplification0.2

    \[\leadsto \sqrt{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))