Average Error: 29.6 → 0.2
Time: 19.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}
double f(double x) {
        double r87206 = x;
        double r87207 = 1.0;
        double r87208 = r87206 + r87207;
        double r87209 = sqrt(r87208);
        double r87210 = sqrt(r87206);
        double r87211 = r87209 - r87210;
        return r87211;
}
↓
double f(double x) {
        double r87212 = 1.0;
        double r87213 = x;
        double r87214 = sqrt(r87213);
        double r87215 = r87213 + r87212;
        double r87216 = sqrt(r87215);
        double r87217 = r87214 + r87216;
        double r87218 = r87212 / r87217;
        double r87219 = sqrt(r87218);
        double r87220 = sqrt(r87212);
        double r87221 = r87219 * r87220;
        double r87222 = sqrt(r87217);
        double r87223 = r87221 / r87222;
        return r87223;
}

Error

Bits error versus x

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Results

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Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.3

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

  9. Applied sqrt-div0.3

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

  10. Applied associate-*r/0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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