Average Error: 29.9 → 0.2
Time: 3.9min
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1 \cdot 1}{1 + 2 \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + x\right)}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1 \cdot 1}{1 + 2 \cdot \left(\sqrt{x + 1} \cdot \sqrt{x} + x\right)}}
double code(double x) {
	return ((double) (((double) sqrt(((double) (x + 1.0)))) - ((double) sqrt(x))));
}
↓
double code(double x) {
	return ((double) sqrt((((double) (1.0 * 1.0)) / ((double) (1.0 + ((double) (2.0 * ((double) (((double) (((double) sqrt(((double) (x + 1.0)))) * ((double) sqrt(x)))) + 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. Using strategy rm

  3. Applied flip--29.7

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied sqrt-unprod0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020181 
(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)))