Average Error: 19.9 → 0.4
Time: 4.5s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x}}{x} \]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x}}{x}

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ (/ (/ 1.0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))) (+ 1.0 x)) x))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	return ((1.0 / (pow(x, -0.5) + pow((1.0 + x), -0.5))) / (1.0 + x)) / x;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.8
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.9

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

  2. Applied egg-rr20.0

    \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

  3. Applied egg-rr19.4

    \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{x + x \cdot x}} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

  4. Applied egg-rr0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x}}{x}} \]

  5. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x}}{x} \]

Reproduce

herbie shell --seed 2022130 
(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)))))