Average Error: 19.9 → 0.4
Time: 39.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
1 \cdot \frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}
double f(double x) {
        double r5795946 = 1.0;
        double r5795947 = x;
        double r5795948 = sqrt(r5795947);
        double r5795949 = r5795946 / r5795948;
        double r5795950 = r5795947 + r5795946;
        double r5795951 = sqrt(r5795950);
        double r5795952 = r5795946 / r5795951;
        double r5795953 = r5795949 - r5795952;
        return r5795953;
}
↓
double f(double x) {
        double r5795954 = 1.0;
        double r5795955 = 1.0;
        double r5795956 = x;
        double r5795957 = r5795956 + r5795954;
        double r5795958 = sqrt(r5795957);
        double r5795959 = r5795955 / r5795958;
        double r5795960 = sqrt(r5795956);
        double r5795961 = r5795959 / r5795960;
        double r5795962 = r5795960 + r5795958;
        double r5795963 = r5795954 * r5795962;
        double r5795964 = r5795961 / r5795963;
        double r5795965 = r5795954 * r5795964;
        return r5795965;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied frac-sub19.9

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

  5. Applied flip--19.7

    \[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot \sqrt{x + 1}\right) \cdot \left(1 \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot 1\right) \cdot \left(\sqrt{x} \cdot 1\right)}{1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  6. Applied associate-/l/19.7

    \[\leadsto \color{blue}{\frac{\left(1 \cdot \sqrt{x + 1}\right) \cdot \left(1 \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot 1\right) \cdot \left(\sqrt{x} \cdot 1\right)}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1\right)}}\]

  7. Taylor expanded around 0 0.8

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

  9. Applied div-inv0.8

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1\right)}}\]

  10. Simplified0.4

    \[\leadsto 1 \cdot \color{blue}{\frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  11. Final simplification0.4

    \[\leadsto 1 \cdot \frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))