Average Error: 29.9 → 0.2
Time: 21.9s
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 r4721512 = x;
        double r4721513 = 1.0;
        double r4721514 = r4721512 + r4721513;
        double r4721515 = sqrt(r4721514);
        double r4721516 = sqrt(r4721512);
        double r4721517 = r4721515 - r4721516;
        return r4721517;
}
↓
double f(double x) {
        double r4721518 = 1.0;
        double r4721519 = x;
        double r4721520 = r4721519 + r4721518;
        double r4721521 = sqrt(r4721520);
        double r4721522 = sqrt(r4721519);
        double r4721523 = r4721521 + r4721522;
        double r4721524 = r4721518 / r4721523;
        return r4721524;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.7

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

  4. Taylor expanded around 0 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 2019121 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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