Average Error: 30.3 → 0.2
Time: 19.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)
double f(double x) {
        double r4894520 = x;
        double r4894521 = 1.0;
        double r4894522 = r4894520 + r4894521;
        double r4894523 = sqrt(r4894522);
        double r4894524 = sqrt(r4894520);
        double r4894525 = r4894523 - r4894524;
        return r4894525;
}
↓
double f(double x) {
        double r4894526 = 1.0;
        double r4894527 = x;
        double r4894528 = r4894527 + r4894526;
        double r4894529 = sqrt(r4894528);
        double r4894530 = sqrt(r4894527);
        double r4894531 = r4894529 + r4894530;
        double r4894532 = r4894526 / r4894531;
        double r4894533 = log1p(r4894532);
        double r4894534 = expm1(r4894533);
        return r4894534;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.3

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--30.1

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

  4. Simplified0.2

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

  6. Applied expm1-log1p-u0.2

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)}\]

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 +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)))