Average Error: 29.7 → 0.2
Time: 14.7s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r84091 = x;
        double r84092 = 1.0;
        double r84093 = r84091 + r84092;
        double r84094 = sqrt(r84093);
        double r84095 = sqrt(r84091);
        double r84096 = r84094 - r84095;
        return r84096;
}
↓
double f(double x) {
        double r84097 = 1.0;
        double r84098 = x;
        double r84099 = r84098 + r84097;
        double r84100 = sqrt(r84099);
        double r84101 = sqrt(r84098);
        double r84102 = r84100 + r84101;
        double r84103 = r84097 / r84102;
        double r84104 = sqrt(r84103);
        double r84105 = sqrt(r84097);
        double r84106 = r84104 * r84105;
        double r84107 = sqrt(r84102);
        double r84108 = r84106 / r84107;
        return r84108;
}

Error

Bits error versus x

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Results

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Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

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

  3. Applied flip--29.5

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

  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 sqrt-div0.3

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

  9. Applied associate-*r/0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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