Average Error: 29.4 → 0.2
Time: 17.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}
double f(double x) {
        double r2975472 = x;
        double r2975473 = 1.0;
        double r2975474 = r2975472 + r2975473;
        double r2975475 = sqrt(r2975474);
        double r2975476 = sqrt(r2975472);
        double r2975477 = r2975475 - r2975476;
        return r2975477;
}
↓
double f(double x) {
        double r2975478 = 1.0;
        double r2975479 = x;
        double r2975480 = sqrt(r2975479);
        double r2975481 = r2975479 + r2975478;
        double r2975482 = sqrt(r2975481);
        double r2975483 = fabs(r2975482);
        double r2975484 = r2975480 + r2975483;
        double r2975485 = r2975478 / r2975484;
        return r2975485;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original29.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

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

  3. Applied flip--29.3

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied rem-sqrt-square0.2

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

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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