Average Error: 29.1 → 0.2
Time: 49.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}
double f(double x) {
        double r8168400 = x;
        double r8168401 = 1.0;
        double r8168402 = r8168400 + r8168401;
        double r8168403 = sqrt(r8168402);
        double r8168404 = sqrt(r8168400);
        double r8168405 = r8168403 - r8168404;
        return r8168405;
}
↓
double f(double x) {
        double r8168406 = 1.0;
        double r8168407 = x;
        double r8168408 = r8168406 + r8168407;
        double r8168409 = sqrt(r8168408);
        double r8168410 = sqrt(r8168407);
        double r8168411 = r8168409 + r8168410;
        double r8168412 = r8168411 * r8168411;
        double r8168413 = -0.5;
        double r8168414 = pow(r8168412, r8168413);
        return r8168414;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original29.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.1

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

  3. Applied flip--28.9

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}\]
  7. Using strategy rm

  8. Applied inv-pow0.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{-1}}}\]

  9. Applied sqrt-pow10.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}}\]

  10. Applied inv-pow0.3

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{-1}}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  11. Applied sqrt-pow10.3

    \[\leadsto \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  12. Applied pow-prod-down0.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(\frac{-1}{2}\right)}}\]

  13. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))