Average Error: 29.9 → 0.2
Time: 5.1s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right) \]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right)

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (log1p (expm1 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
↓
double code(double x) {
	return log1p(expm1((1.0 / (sqrt((1.0 + x)) + sqrt(x)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 29.9

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

  2. Applied flip--_binary6429.7

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

  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

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

  6. Applied fma-def_binary640.2

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

  7. Applied log1p-expm1-u_binary640.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt{1 + x}}, \sqrt{\sqrt{1 + x}}, \sqrt{x}\right)}\right)\right)} \]

  8. Simplified0.2

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)}\right) \]

  9. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right) \]

Reproduce

herbie shell --seed 2022125 
(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)))