Average Error: 19.5 → 0.3
Time: 1.5m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + 1}}
double f(double x) {
        double r4294979 = 1.0;
        double r4294980 = x;
        double r4294981 = sqrt(r4294980);
        double r4294982 = r4294979 / r4294981;
        double r4294983 = r4294980 + r4294979;
        double r4294984 = sqrt(r4294983);
        double r4294985 = r4294979 / r4294984;
        double r4294986 = r4294982 - r4294985;
        return r4294986;
}
↓
double f(double x) {
        double r4294987 = 1.0;
        double r4294988 = x;
        double r4294989 = r4294988 + r4294987;
        double r4294990 = sqrt(r4294989);
        double r4294991 = sqrt(r4294988);
        double r4294992 = r4294990 + r4294991;
        double r4294993 = r4294987 / r4294992;
        double r4294994 = r4294993 / r4294988;
        double r4294995 = sqrt(r4294994);
        double r4294996 = sqrt(r4294993);
        double r4294997 = r4294996 / r4294990;
        double r4294998 = r4294995 * r4294997;
        return r4294998;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied frac-sub19.4

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

  4. Simplified19.4

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

  6. Applied flip--19.2

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

  7. Simplified0.4

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied times-frac0.4

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

  12. Applied sqrt-undiv0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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