Average Error: 19.7 → 19.7
Time: 37.0s
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 r4213566 = 1.0;
        double r4213567 = x;
        double r4213568 = sqrt(r4213567);
        double r4213569 = r4213566 / r4213568;
        double r4213570 = r4213567 + r4213566;
        double r4213571 = sqrt(r4213570);
        double r4213572 = r4213566 / r4213571;
        double r4213573 = r4213569 - r4213572;
        return r4213573;
}
↓
double f(double x) {
        double r4213574 = 1.0;
        double r4213575 = x;
        double r4213576 = sqrt(r4213575);
        double r4213577 = r4213574 / r4213576;
        double r4213578 = hypot(r4213576, r4213574);
        double r4213579 = r4213574 / r4213578;
        double r4213580 = r4213577 - r4213579;
        return r4213580;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original19.7
Target0.8
Herbie19.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied *-un-lft-identity19.7

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

  4. Applied add-sqr-sqrt19.7

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

  5. Applied hypot-def19.7

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

  6. Final simplification19.7

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

Reproduce

herbie shell --seed 2019144 +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)))))