Average Error: 19.8 → 19.8
Time: 54.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} - \frac{1}{\mathsf{hypot}\left(\sqrt{x}, \sqrt{1}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} - \frac{1}{\mathsf{hypot}\left(\sqrt{x}, \sqrt{1}\right)}
double f(double x) {
        double r334819 = 1.0;
        double r334820 = x;
        double r334821 = sqrt(r334820);
        double r334822 = r334819 / r334821;
        double r334823 = r334820 + r334819;
        double r334824 = sqrt(r334823);
        double r334825 = r334819 / r334824;
        double r334826 = r334822 - r334825;
        return r334826;
}
↓
double f(double x) {
        double r334827 = 1.0;
        double r334828 = x;
        double r334829 = sqrt(r334828);
        double r334830 = r334827 / r334829;
        double r334831 = sqrt(r334827);
        double r334832 = hypot(r334829, r334831);
        double r334833 = r334827 / r334832;
        double r334834 = r334830 - r334833;
        return r334834;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 19.8

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

  3. Applied add-sqr-sqrt19.8

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

  4. Applied add-sqr-sqrt19.8

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

  5. Applied hypot-def19.8

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

  6. Final simplification19.8

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

Reproduce

herbie shell --seed 2019199 +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)))))