Average Error: 30.0 → 0.2
Time: 33.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r3458301 = x;
        double r3458302 = 1.0;
        double r3458303 = r3458301 + r3458302;
        double r3458304 = sqrt(r3458303);
        double r3458305 = sqrt(r3458301);
        double r3458306 = r3458304 - r3458305;
        return r3458306;
}
↓
double f(double x) {
        double r3458307 = 1.0;
        double r3458308 = x;
        double r3458309 = r3458308 + r3458307;
        double r3458310 = sqrt(r3458309);
        double r3458311 = sqrt(r3458308);
        double r3458312 = r3458310 + r3458311;
        double r3458313 = r3458307 / r3458312;
        return r3458313;
}

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. Using strategy rm

  3. Applied flip--29.8

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

  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-div0.3

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

  9. Applied associate-*r/0.2

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

  11. Applied sqrt-div0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied associate-/l/0.3

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))