Average Error: 30.2 → 0.2
Time: 12.6s
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 r400252 = x;
        double r400253 = 1.0;
        double r400254 = r400252 + r400253;
        double r400255 = sqrt(r400254);
        double r400256 = sqrt(r400252);
        double r400257 = r400255 - r400256;
        return r400257;
}
↓
double f(double x) {
        double r400258 = 1.0;
        double r400259 = x;
        double r400260 = r400259 + r400258;
        double r400261 = sqrt(r400260);
        double r400262 = sqrt(r400259);
        double r400263 = r400261 + r400262;
        double r400264 = r400258 / r400263;
        return r400264;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.0

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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