Average Error: 30.0 → 0.2
Time: 4.6s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1} \]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1}

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

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 flip--_binary6429.8

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

  3. Simplified0.2

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

  4. Simplified0.2

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

  5. Applied add-sqr-sqrt_binary640.3

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

  6. Applied pow1_binary640.3

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

  7. Applied pow-flip_binary640.3

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

  8. Applied sqrt-pow1_binary640.3

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

  9. Applied pow1_binary640.3

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

  10. Applied pow-flip_binary640.3

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

  11. Applied sqrt-pow1_binary640.3

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

  12. Applied pow-prod-up_binary640.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2022024 
(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)))