Average Error: 30.3 → 0.2
Time: 15.9s
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 r23723992 = x;
        double r23723993 = 1.0;
        double r23723994 = r23723992 + r23723993;
        double r23723995 = sqrt(r23723994);
        double r23723996 = sqrt(r23723992);
        double r23723997 = r23723995 - r23723996;
        return r23723997;
}
↓
double f(double x) {
        double r23723998 = 1.0;
        double r23723999 = x;
        double r23724000 = r23723999 + r23723998;
        double r23724001 = sqrt(r23724000);
        double r23724002 = sqrt(r23723999);
        double r23724003 = r23724001 + r23724002;
        double r23724004 = r23723998 / r23724003;
        return r23724004;
}

Error

Bits error versus x

Try it out

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Results

Enter valid numbers for all inputs

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. Simplified29.7

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

  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 2019171 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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