Average Error: 19.8 → 19.8
Time: 25.8s
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 r3621777 = 1.0;
        double r3621778 = x;
        double r3621779 = sqrt(r3621778);
        double r3621780 = r3621777 / r3621779;
        double r3621781 = r3621778 + r3621777;
        double r3621782 = sqrt(r3621781);
        double r3621783 = r3621777 / r3621782;
        double r3621784 = r3621780 - r3621783;
        return r3621784;
}
↓
double f(double x) {
        double r3621785 = 1.0;
        double r3621786 = x;
        double r3621787 = sqrt(r3621786);
        double r3621788 = r3621785 / r3621787;
        double r3621789 = hypot(r3621787, r3621785);
        double r3621790 = r3621785 / r3621789;
        double r3621791 = r3621788 - r3621790;
        return r3621791;
}

Error

Bits error versus x

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Results

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Target

Original19.8
Target0.7
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 *-un-lft-identity19.8

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

  4. Applied add-sqr-sqrt19.8

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

  5. Applied hypot-def19.8

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

  6. Final simplification19.8

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

Reproduce

herbie shell --seed 2019135 +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)))))