Average Error: 30.5 → 0.2
Time: 4.2s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\sqrt{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}} \]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}}

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 30.5

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

  3. Applied flip--_binary6430.3

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt_binary640.3

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

  8. Applied sqrt-unprod_binary640.2

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

  9. Simplified0.2

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}}} \]
  10. Using strategy rm

  11. Applied *-un-lft-identity_binary640.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2021211 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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