Average Error: 29.3 → 0.2
Time: 16.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r95215 = x;
        double r95216 = 1.0;
        double r95217 = r95215 + r95216;
        double r95218 = sqrt(r95217);
        double r95219 = sqrt(r95215);
        double r95220 = r95218 - r95219;
        return r95220;
}
↓
double f(double x) {
        double r95221 = 1.0;
        double r95222 = 1.0;
        double r95223 = x;
        double r95224 = r95223 + r95221;
        double r95225 = sqrt(r95224);
        double r95226 = sqrt(r95223);
        double r95227 = r95225 + r95226;
        double r95228 = r95222 / r95227;
        double r95229 = r95221 * r95228;
        return r95229;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.2

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

  9. Applied div-inv0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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