Average Error: 29.6 → 0.2
Time: 13.7s
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 r24694991 = x;
        double r24694992 = 1.0;
        double r24694993 = r24694991 + r24694992;
        double r24694994 = sqrt(r24694993);
        double r24694995 = sqrt(r24694991);
        double r24694996 = r24694994 - r24694995;
        return r24694996;
}
↓
double f(double x) {
        double r24694997 = 1.0;
        double r24694998 = x;
        double r24694999 = r24694998 + r24694997;
        double r24695000 = sqrt(r24694999);
        double r24695001 = sqrt(r24694998);
        double r24695002 = r24695000 + r24695001;
        double r24695003 = r24694997 / r24695002;
        return r24695003;
}

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. Simplified29.0

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

  5. Taylor expanded around 0 0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))