Average Error: 19.6 → 19.5
Time: 21.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} - \frac{1}{\mathsf{hypot}\left(\sqrt{x}, 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} - \frac{1}{\mathsf{hypot}\left(\sqrt{x}, 1\right)}
double f(double x) {
        double r5047961 = 1.0;
        double r5047962 = x;
        double r5047963 = sqrt(r5047962);
        double r5047964 = r5047961 / r5047963;
        double r5047965 = r5047962 + r5047961;
        double r5047966 = sqrt(r5047965);
        double r5047967 = r5047961 / r5047966;
        double r5047968 = r5047964 - r5047967;
        return r5047968;
}
↓
double f(double x) {
        double r5047969 = 1.0;
        double r5047970 = x;
        double r5047971 = sqrt(r5047970);
        double r5047972 = r5047969 / r5047971;
        double r5047973 = hypot(r5047971, r5047969);
        double r5047974 = r5047969 / r5047973;
        double r5047975 = r5047972 - r5047974;
        return r5047975;
}

Error

Bits error versus x

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Results

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Target

Original19.6
Target0.7
Herbie19.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied *-un-lft-identity19.6

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

  4. Applied add-sqr-sqrt19.5

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

  5. Applied hypot-def19.5

    \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\mathsf{hypot}\left(\sqrt{x}, 1\right)}}\]

  6. Final simplification19.5

    \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\mathsf{hypot}\left(\sqrt{x}, 1\right)}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(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)))))