Average Error: 29.3 → 0.2
Time: 17.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r4242636 = x;
        double r4242637 = 1.0;
        double r4242638 = r4242636 + r4242637;
        double r4242639 = sqrt(r4242638);
        double r4242640 = sqrt(r4242636);
        double r4242641 = r4242639 - r4242640;
        return r4242641;
}
↓
double f(double x) {
        double r4242642 = 1.0;
        double r4242643 = x;
        double r4242644 = r4242643 + r4242642;
        double r4242645 = sqrt(r4242644);
        double r4242646 = sqrt(r4242643);
        double r4242647 = r4242645 + r4242646;
        double r4242648 = r4242642 / r4242647;
        return r4242648;
}

Error

Bits error versus x

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Results

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Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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