Average Error: 19.7 → 19.7
Time: 35.6s
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 r3414096 = 1.0;
        double r3414097 = x;
        double r3414098 = sqrt(r3414097);
        double r3414099 = r3414096 / r3414098;
        double r3414100 = r3414097 + r3414096;
        double r3414101 = sqrt(r3414100);
        double r3414102 = r3414096 / r3414101;
        double r3414103 = r3414099 - r3414102;
        return r3414103;
}
↓
double f(double x) {
        double r3414104 = 1.0;
        double r3414105 = x;
        double r3414106 = sqrt(r3414105);
        double r3414107 = r3414104 / r3414106;
        double r3414108 = hypot(r3414106, r3414104);
        double r3414109 = r3414104 / r3414108;
        double r3414110 = r3414107 - r3414109;
        return r3414110;
}

Error

Bits error versus x

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Results

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Target

Original19.7
Target0.7
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 2019143 +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)))))