Average Error: 30.1 → 0.2
Time: 1.0m
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 r10977419 = x;
        double r10977420 = 1.0;
        double r10977421 = r10977419 + r10977420;
        double r10977422 = sqrt(r10977421);
        double r10977423 = sqrt(r10977419);
        double r10977424 = r10977422 - r10977423;
        return r10977424;
}
↓
double f(double x) {
        double r10977425 = 1.0;
        double r10977426 = x;
        double r10977427 = r10977426 + r10977425;
        double r10977428 = sqrt(r10977427);
        double r10977429 = sqrt(r10977426);
        double r10977430 = r10977428 + r10977429;
        double r10977431 = r10977425 / r10977430;
        return r10977431;
}

Error

Bits error versus x

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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. Using strategy rm

  3. Applied flip--29.9

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

  4. Taylor expanded around inf 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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