Average Error: 30.3 → 0.2
Time: 16.3s
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 r27934099 = x;
        double r27934100 = 1.0;
        double r27934101 = r27934099 + r27934100;
        double r27934102 = sqrt(r27934101);
        double r27934103 = sqrt(r27934099);
        double r27934104 = r27934102 - r27934103;
        return r27934104;
}
↓
double f(double x) {
        double r27934105 = 1.0;
        double r27934106 = x;
        double r27934107 = r27934106 + r27934105;
        double r27934108 = sqrt(r27934107);
        double r27934109 = sqrt(r27934106);
        double r27934110 = r27934108 + r27934109;
        double r27934111 = r27934105 / r27934110;
        return r27934111;
}

Error

Bits error versus x

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Results

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Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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