Average Error: 30.0 → 0.3
Time: 12.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r101670 = x;
        double r101671 = 1.0;
        double r101672 = r101670 + r101671;
        double r101673 = sqrt(r101672);
        double r101674 = sqrt(r101670);
        double r101675 = r101673 - r101674;
        return r101675;
}
↓
double f(double x) {
        double r101676 = 1.0;
        double r101677 = x;
        double r101678 = sqrt(r101677);
        double r101679 = r101677 + r101676;
        double r101680 = sqrt(r101679);
        double r101681 = sqrt(r101680);
        double r101682 = r101681 * r101681;
        double r101683 = r101678 + r101682;
        double r101684 = r101676 / r101683;
        return r101684;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

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

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1 + 0}{\sqrt{x} + \sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}\]

  8. Applied sqrt-prod0.3

    \[\leadsto \frac{1 + 0}{\sqrt{x} + \color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}}\]

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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