Average Error: 30.2 → 0.2
Time: 8.0s
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 r372391 = x;
        double r372392 = 1.0;
        double r372393 = r372391 + r372392;
        double r372394 = sqrt(r372393);
        double r372395 = sqrt(r372391);
        double r372396 = r372394 - r372395;
        return r372396;
}
↓
double f(double x) {
        double r372397 = 1.0;
        double r372398 = x;
        double r372399 = r372398 + r372397;
        double r372400 = sqrt(r372399);
        double r372401 = sqrt(r372398);
        double r372402 = r372400 + r372401;
        double r372403 = r372397 / r372402;
        return r372403;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied times-frac0.3

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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