Average Error: 30.1 → 0.2
Time: 13.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}
double f(double x) {
        double r122876 = x;
        double r122877 = 1.0;
        double r122878 = r122876 + r122877;
        double r122879 = sqrt(r122878);
        double r122880 = sqrt(r122876);
        double r122881 = r122879 - r122880;
        return r122881;
}
↓
double f(double x) {
        double r122882 = 1.0;
        double r122883 = x;
        double r122884 = sqrt(r122883);
        double r122885 = r122883 + r122882;
        double r122886 = sqrt(r122885);
        double r122887 = fabs(r122886);
        double r122888 = r122884 + r122887;
        double r122889 = r122882 / r122888;
        return r122889;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.1

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

  2. Simplified30.1

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

  4. Applied flip--29.8

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied rem-sqrt-square0.2

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

  9. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))